SENSORLESS VECTOR CONTROL OF INDUCTION MOTOR ...
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SENSORLESS VECTOR CONTROL OF INDUCTION MOTOR BASED ON FLUX AND SPEED ESTIMATION
A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF MIDDLE EAST TECHNICAL UNIVERSITY
BY
BARIŞ TUĞRUL ERTUĞRUL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF MASTER OF SCIENCE IN
ELECTRICAL AND ELECTRONICS ENGINEERING
DECEMBER 2008
Approval of the thesis:
SENSORLESS VECTOR CONTROL OF INDUCTION MOTOR BASED ON
FLUX AND SPEED ESTIMATION
submitted by BARIŞ TUĞRUL ERTUĞRUL in partial fulfilment of the requirements for the degree of Master of Science in Electrical and Electronics Engineering, Middle East Technical University by,
Prof. Dr. Canan ÖZGEN Dean, Graduate School of Natural and Applied Sciences ______________ Prof. Dr. İsmet ERKMEN Head of Department, Electrical and Electronics Engineering ______________ Prof. Dr. Aydın ERSAK Supervisor, Electrical and Electronics Engineering, METU ______________ Examining Committee Members: Prof. Dr. Muammer ERMİŞ Electrical and Electronics Engineering, METU ______________ Prof. Dr. Aydın ERSAK Electrical and Electronics Engineering, METU ______________ Prof. Dr. Işık ÇADIRCI Electrical and Electronics Engineering, Hacettepe Univ. ______________ Assist. Prof. Dr. Ahmet M. HAVA Electrical and Electronics Engineering, METU ______________ Assist. Prof. Dr. M. Timur AYDEMİR Electrical and Electronics Engineering, Gazi University ______________
Date: 02.12.2008
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I hereby declare that all information in this document has been obtained and
presented in accordance with academic rules and ethical conduct. I also declare
that, as required by these rules and conduct, I have fully cited and referenced
all material and results that are not original to this work.
Name, Last name : Barış Tuğrul ERTUĞRUL
Signature :
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ABSTRACT
SENSORLESS VECTOR CONTROL OF INDUCTION MOTOR BASED ON FLUX AND SPEED ESTIMATION
ERTUĞRUL, Barış Tuğrul
M. Sc. Department of Electrical and Electronics Engineering
Supervisor: Prof. Dr. Aydın Ersak
December 2008, 148 pages
The main focus of the study is the implementation of techniques regarding flux
estimation and rotor speed estimation by the use of sensorless closed-loop observers.
Within this framework, the information about the mathematical representation of the
induction motor, pulse width modulation technique and flux oriented vector control
techniques together with speed adaptive flux estimation –a kind of sensorless closed
loop estimation technique- and Kalman filters is given.
With the comparison of sensorless closed-loop speed estimation techniques, it
has been attempted to identify their superiority and inferiority to each other by the
use of simulation models and real-time experiments. In the experiments, the
performance of the techniques developed and used in the thesis has been examined
under extensively changing speed and load conditions. The real-time experiments
have been carried out by the use of TI TMS320F2812 digital signal processor,
XILINX XCS2S150E Field Programmable Gate Array (FPGA), control card and the
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motor drive card Furthermore, Matlab “Embedded Target for the TI C2000 DSP”
and “Code Composer Studio” software tools have been used.
The simulations and experiments conducted in the study have illustrated that it is
possible to increase the performance at low speeds at the expense of increased
computational burden on the processor. However, in order to control the motor at
zero speed, high frequency signal implementation should be used as well as a
different electronic hardware.
Key words: Speed control of induction machine, sensorless closed loop field
oriented control, flux observer, speed observer
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ÖZ
HIZ DUYAÇSIZ ENDÜKSİYON MOTORUNUN AKI VE HIZ KESTİRİM
YÖNTEMLERİNE DAYALI VEKTÖR DENETİMİ
ERTUĞRUL, Barış Tuğrul
Yüksek Lisans, Elektrik ve Elektronik Mühendisliği Bölümü
Tez Yöneticisi: Prof. Dr. Aydın Ersak
Aralık 2008, 148 sayfa
Bu çalışma ensüksiyon motorlarını esas alan hız kontrollü motor sürücü tasarımını
ve uygulamalarını kapsamaktadır. Bu çalışmanın temel olarak yoğunlaştığı alan hız-
duyaçsız kapalı döngü gözleyiciler kullanılarak manyetik akı kestirimi ve rotor hızı
kestirim tekniklerinin uygulanmasıdır. Bu çerçevede tezde endüksiyon motorunun
matematiksel modellenmesi, darbe genliği modülasyonu tekniği ve akı yönlendirmeli
vektör kontrol teknikleriyle beraber duyaçsız kapalı döngü kestirim tekniklerinden
hız uyarlamalı manyetik akı kestirim metodu ile Kalman filtreler hakkında bilgi
verilmiştir.
Duyaçsız kapalı döngü hız kestirim yöntemlerinin birbirlerine göre olan
üstünlükleri ile zayıflıkları benzetim modelleri ve gerçek zamanlı deneylerle ortaya
konmaya çalışılmıştır. Deneylerde yöntemlerin geniş bir hız bandında ve yük
altındaki performansı incelenmiştir.
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Gerçek zamanlı deneyler TI TMS320F2812 sayısal işaret işlemcisi, XILINX
XCS2S150E alanda programlanabilir kapı dizileri (FPGA) ile birlikte çeşitli
analogdan sayısala, sayısaldan analoga çevirimleri sağlayan yongalar ve çevre
elemanlardan oluşan kontrol kartı ile birlikte temel olarak güç anahtarlama, işaret
arayüz uyumlama, gerilim ve akım ölçme devrelerini içeren motor sürücü kartı
vasıtasıyla yapılmıştır. Ayrıca, yazılım arayüzü olarak Matlab “Embedded Target for
the TI C2000 DSP” ve “Code Composer Studio” yazılım araçları kullanılmıştır.
Çalışma süresince ortaya konan benzetim ve deneyler göstermiştir ki işlemci
yükünü arttırmak suretiyle düşük hızlarda performansı arttırmak mümkün olmaktadır
ancak sıfır hızda motor kontrolünü gerçekleştirmek için farklı bir elektronik
donanımla birlikte yüksek frekans işaret uygulama yöntemleri kullanılmalıdır.
Anahtar Kelimeler: Endüksiyon makinelerinin hız denetimi, duyaçsız kapalı-
döngü alan yönlendirmeli denetim, akı gözleyici, hız gözleyici
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ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my supervisor Prof. Dr. Aydın
Ersak for his encouragement and valuable supervision throughout the study.
I would like to thank to ASELSAN Inc. for the facilities provided and my
colleagues for their support during the course of the thesis.
Thanks a lot to my friends, Ömer GÖKSU, Evrim Onur ARI, Murat ERTEK,
Günay ŞİMŞEK and Doğan YILDIRIM for their help during the experimental stage
of this work.
I appreciate my family due to their great trust.
The last but not least acknowledgment is to my precious fiancée Didem
SÜLÜKÇÜ for her encouragement and continuous emotional support.
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TABLE OF CONTENTS
ABSTRACT................................................................................................................................... ......iv
ÖZ .................................................................................................................................................... vi
ACKNOWLEDGEMENTS.............................................................................................................. viii
TABLE OF CONTENTS.................................................................................................................... ix
LIST OF TABLES .............................................................................................................................. xi
LIST OF FIGURES ......................................................................................................................... ..xii
LIST OF SYMBOLS..................................................................................................................... ..xviii
CHAPTERS
1 INTRODUCTION ...................................................................................................................... 1 1.1 INDUCTION MACHINE DRIVES ....................................................................................... 1 1.2 THE FIELD ORIENTED CONTROL (VECTOR CONTROL) OF INDUCTION MACHINES ......... 1 1.3 INDUCTION MACHINE FLUX OBSERVATION ................................................................... 2 1.4 SENSORLESS VECTOR CONTROL OF INDUCTION MACHINE ............................................ 5 1.5 STRUCTURE OF THE CHAPTERS ...................................................................................... 7
2 INDUCTION MACHINE MODELING, FIELD ORIENTED CONTROL AND PWM WITH SPACE VECTOR THEORY ....................................................................................... 8
2.1 SYSTEM EQUATIONS IN THE STATIONARY A,B,C REFERENCE FRAME ............................ 8 2.1.1. Determination of Induction Machine Inductances [21] ............................................... 11 2.1.2. Three-Phase to Two-Phase Transformations ............................................................... 15
2.1.2.1. The Clarke Transformation [1] .......................................................................................... 15 2.1.2.2. The Park Transformation [1].............................................................................................. 17
2.2 REFERENCE FRAMES.................................................................................................... 19 2.2.1. Induction Motor Model in the Arbitrary dq0 Reference Frame ................................... 19 2.2.2. Induction Motor Model in dq0 Stationary and Synchronous Reference Frames.......... 22
2.3 FIELD ORIENTED CONTROL (FOC) .............................................................................. 25 2.4 SPACE VECTOR PULSE WIDTH MODULATION (SVPWM) ............................................ 31
2.4.1. Voltage Fed Inverter (VSI) ........................................................................................... 31 2.4.2. Voltage Space Vectors.................................................................................................. 33 2.4.3. SVPWM Application to the Static Power Bridge.......................................................... 35
3 OBSERVERS FOR SENSORLESS FIELD ORIENTED CONTROL OF INDUCTION MACHINE ............................................................................................................................... 45
3.1. SPEED ADAPTIVE FLUX OBSERVER FOR INDUCTION MOTOR ....................................... 47 3.1.1. Flux Estimation Based on the Induction Motor Model ................................................ 47
3.1.1.1. Estimation of Rotor Flux Angle ......................................................................................... 50 3.1.2. Adaptive Scheme for Speed Estimation ........................................................................ 51
3.2. KALMAN FILTER FOR SPEED ESTIMATION ................................................................... 56 3.2.1. Discrete Kalman Filter................................................................................................. 56 3.2.2. “Obtaining “Synchronous Speed, ws” Speed Data through the use of “Rotor Flux
Angle” ......................................................................................................................... 61 3.2.3. Extended Kalman Filter (EKF) .................................................................................... 62
3.2.3.1. Selection of the Time-Domain Machine Model ................................................................. 63 3.2.3.2. Discretization of the Induction Motor Model..................................................................... 66
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3.2.3.3. Determination of the Noise and State Covariance Matrices Q, R, P .................................. 67 3.2.3.4. Implementation of the Discretized EKF Algorithm; Tuning.............................................. 68
4 SIMULATIONS AND EXPERIMENTAL WORK............................................................... 75 4.1 EXPERIMENTAL WORK ................................................................................................ 75
4.1.1 Induction Motor Data................................................................................................... 75 4.1.2. Experimental Set-up ..................................................................................................... 79 4.1.3 Experimental Results of Speed Adaptive Flux Observer .............................................. 83
4.1.3.1 No-Load Experiments of Speed Adaptive Flux Observer .................................................. 84 4.1.3.2. The Speed Estimator Performance under Switched Loading ............................................. 89 4.1.3.3. The Speed Estimator Performance under Accelerating Load............................................. 97 4.1.3.4. The Speed Estimator Performance under No-Load Speed Reversal ................................ 100
4.1.4 Experimental Results of Kalman Filter for Speed Estimation.................................... 103 4.1.4.1. No-Load Experiments of Kalman Filter for Speed Estimation ........................................ 103 4.1.4.2. The Kalman Filter for Speed Estimation Performance under Switched Loading............. 110 4.1.4.3. The Kalman Filter for Speed Estimation Performance under Accelerating Load ............ 116 4.1.4.4. The Kalman Filter for Speed Estimation Performance under No-Load Speed Reversal .. 120
4.1.5 Experimental Results of Parallel Run of Speed Adaptive Flux Observer and Kalman Filter for Speed Estimation ....................................................................................... 123
4.1.5.1. No-Load Experiments of Parallel Run of Speed Adaptive Flux Observer and Kalman Filter for Speed Estimation........................................................................................................ 123
4.1.5.2. Parallel Run of Speed Adaptive Flux Observer and Kalman Filter for Speed Estimation under Switched Loading .................................................................................................. 125
4.1.5.3. Parallel Run of Speed Adaptive Flux Observer and Kalman Filter for Speed Estimation under Accelerating Load .................................................................................................. 129
4.1.5.4. Parallel Run of Speed Adaptive Flux Observer and Kalman Filter for Speed Estimation under No-load Speed Reversal ......................................................................................... 132
4.2 SIMULATIONS OF EXTENDED KALMAN FILTER .......................................................... 134 4.2.1. Tuning of EKF Covariance Matrices ......................................................................... 135 4.2.2. Simulations at Lower Speeds...................................................................................... 135
5 CONCLUSION ....................................................................................................................... 143 6 REFERENCES ....................................................................................................................... 146
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LIST OF TABLES
Table 2-1 Instantaneous Basic Voltage Vectors [25]................................................. 33
Table 2-2 Power bridge output voltages (Van, Vbn, Vcn) ........................................... 35
Table 2-3 Stator voltages in (dsqs) frame and related voltage vector......................... 36
Table 2-4 Durations of sector boundary..................................................................... 40
Table 2-5 Assigned duty cycles to the PWM outputs ................................................ 41
Table 4-1 Induction motor electrical data .................................................................. 75
Table 4-2 Induction motor parameters....................................................................... 76
Table 4-3 Direct on-line starting per unit definitions................................................. 76
Table 4-4 Control parameters used at the speed adaptive flux observer experiments84
Table 4-5 Loading measurements .............................................................................. 90
Table 4-6 Control parameters used at Kalman filter for speed estimation experiments
............................................................................................................. 103
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LIST OF FIGURES
Figure 1-1 Inputs and outputs of the voltage model flux observer (VMFO)............... 3
Figure 1-2 Inputs and outputs of the current model flux observer............................... 4
Figure 1-3 Inputs and outputs of the speed adaptive flux observer ............................. 5
Figure 2-1 Axial view of an induction machine........................................................... 9
Figure 2-2 Magnetic axes of three phase induction machine....................................... 9
Figure 2-3 Relationship between the α, β and the abc quantities [23] ....................... 16
Figure 2-4 Relationship between the dq and the abc quantities [23] ......................... 18
Figure 2-5 The Park and the Clarke transformations at the machine side [1]............ 27
Figure 2-6 Variable transformation in the field oriented control [1] ......................... 28
Figure 2-7 Phasor diagram of the field oriented drive system................................... 29
Figure 2-8 Indirect field oriented drive system.......................................................... 29
Figure 2-9 Direct field oriented drive system ............................................................ 30
Figure 2-10 Indirect field oriented induction motor drive system with sensor [1] .... 30
Figure 2-11 The block diagram of VSI supplied from a diode rectifier .................... 32
Figure 2-12 Three phase voltage source inverter supplying induction motor [1]...... 32
Figure 2-13 Basic space vectors [25] ......................................................................... 34
Figure 2-14 Projection of the reference voltage vector............................................. 38
Figure 3-1 The block diagram of rotor flux angle estimation.................................... 51
Figure 3-2 Adaptive state observer ............................................................................ 52
Figure 3-3 Discrete Kalman filter algorithm.............................................................. 59
Figure 3-4 The structure of EKF algorithm ............................................................... 70
Figure 4-1 The direct on-line starting moment vs motor speed graph....................... 77
Figure 4-2 The direct on-line starting motor current vs motor speed graph .............. 78
Figure 4-3 The direct on-line starting time vs motor speed graph............................. 78
Figure 4-4 The schematic block diagrams of experimental set-up ............................ 79
Figure 4-5 The Magtrol test bench load response...................................................... 82
Figure 4-6 The experimental set-up ........................................................................... 83
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Figure 4-7 50 rpm speed reference, motor speed estimate......................................... 85
Figure 4-8 100 rpm speed reference, motor speed estimate....................................... 85
Figure 4-9 500 rpm speed reference, motor speed estimate....................................... 86
Figure 4-10 1000 rpm speed reference, motor speed estimate................................... 87
Figure 4-11 1500 rpm speed reference, motor speed estimate................................... 87
Figure 4-12 50 rpm speed reference, motor quadrature encoder position ................. 88
Figure 4-13 50 rpm speed reference, motor phase currents ....................................... 88
Figure 4-14 50 rpm speed reference, motor phase voltages....................................... 89
Figure 4-15 50 rpm speed reference, motor speed estimate under switched loading-1
............................................................................................................... 91
Figure 4-16 50 rpm speed reference, motor speed estimate under switched loading-2
............................................................................................................... 91
Figure 4-17 100 rpm speed reference, motor speed estimate under switched loading-1
............................................................................................................... 92
Figure 4-18 100 rpm speed reference, motor speed estimate under switched loading-2
............................................................................................................... 93
Figure 4-19 500 rpm speed reference, motor speed estimate under switched loading-1
............................................................................................................... 93
Figure 4-20 500 rpm speed reference, motor speed estimate under switched loading-2
............................................................................................................... 94
Figure 4-21 1000 rpm speed reference, motor speed estimate under switched loading-
1............................................................................................................. 94
Figure 4-22 1000 rpm speed reference, motor speed estimate under switched loading-
2............................................................................................................. 95
Figure 4-23 1500 rpm speed reference, motor speed estimate under switched loading-
1............................................................................................................. 95
Figure 4-24 1500 rpm speed reference, motor speed estimate under switched loading-
2............................................................................................................. 96
Figure 4-25 50 rpm to 500 rpm speed reference, motor speed estimate under
accelerating load-1 ................................................................................ 97
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Figure 4-26 50 rpm to 500 rpm speed reference, motor speed estimate under
accelerating load-2 ................................................................................ 98
Figure 4-27 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-1 ................................................................................ 98
Figure 4-28 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-2 ................................................................................ 99
Figure 4-29 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-1 ................................................................................ 99
Figure 4-30 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-2 .............................................................................. 100
Figure 4-31 The speed reference, motor speed estimate under no-load speed reversal
for the speed range 50 rpm to -50 rpm................................................ 101
Figure 4-32 The speed reference, motor speed estimate under no-load speed reversal
for the speed range 500 rpm to -500 rpm............................................ 101
Figure 4-33 The speed reference, motor speed estimate under no-load speed reversal
for the speed range 1000 rpm to -1000 rpm........................................ 102
Figure 4-34 50 rpm speed reference, motor speed estimate..................................... 105
Figure 4-35 100 rpm speed reference, motor speed estimate................................... 105
Figure 4-36 250 rpm speed reference, motor speed estimate................................... 106
Figure 4-37 500 rpm speed reference, motor speed estimate................................... 107
Figure 4-38 1000 rpm speed reference, motor speed estimate................................. 107
Figure 4-39 1500 rpm speed reference, motor speed estimate................................. 108
Figure 4-40 50rpm speed reference, motor quadrature encoder position ................ 109
Figure 4-41 50 rpm speed reference, motor phase currents ..................................... 109
Figure 4-42 50 rpm speed reference, motor phase voltages..................................... 110
Figure 4-43 50 rpm speed reference, motor speed estimate under switched loading-1
............................................................................................................. 111
Figure 4-44 50 rpm speed reference, motor speed estimate under switched loading-2
............................................................................................................. 111
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Figure 4-45 100 rpm speed reference, motor speed estimate under switched loading-1
............................................................................................................. 112
Figure 4-46 100 rpm speed reference, motor speed estimate under switched loading-2
............................................................................................................. 113
Figure 4-47 500 rpm speed reference, motor speed estimate under switched loading-1
............................................................................................................. 113
Figure 4-48 500 rpm speed reference, motor speed estimate under switched loading-2
............................................................................................................. 114
Figure 4-49 1000 rpm speed reference, motor speed estimate under switched loading-
1........................................................................................................... 114
Figure 4-50 1000 rpm speed reference, motor speed estimate under switched loading-
2........................................................................................................... 115
Figure 4-51 1500 rpm speed reference, motor speed estimate under switched loading-
1........................................................................................................... 115
Figure 4-52 1500 rpm speed reference, motor speed estimate under switched loading-
2........................................................................................................... 116
Figure 4-53 100 rpm to 250 rpm speed reference, motor speed estimate under
accelerating load-1 .............................................................................. 117
Figure 4-54 100 rpm to 250 rpm speed reference, motor speed estimate under
accelerating load-2 .............................................................................. 118
Figure 4-55 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-1 .............................................................................. 118
Figure 4-56 500rpm to 750rpm speed reference, motor speed estimate under
accelerating load-2 .............................................................................. 119
Figure 4-57 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-1 .............................................................................. 119
Figure 4-58 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-2 .............................................................................. 120
Figure 4-59 100 rpm to -100 rpm speed reference, motor speed estimate under no-
load speed reversal .............................................................................. 121
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Figure 4-60 500 rpm to - 500 rpm speed reference, motor speed estimate under no-
load speed reversal .............................................................................. 121
Figure 4-61 1000 rpm to -1000 rpm speed reference, motor speed estimate under no-
load speed reversal .............................................................................. 122
Figure 4-62 500 rpm speed reference, motor speed estimate................................... 124
Figure 4-63 1000 rpm speed reference, motor speed estimate................................. 124
Figure 4-64 1500 rpm speed reference, motor speed estimate................................. 125
Figure 4-65 500 rpm speed reference, motor speed estimate under switched loading-1
............................................................................................................. 126
Figure 4-66 500 rpm speed reference, motor speed estimate under switched loading-2
............................................................................................................. 127
Figure 4-67 1000 rpm speed reference, motor speed estimate under switched loading-
1........................................................................................................... 127
Figure 4-68 1000 rpm speed reference, motor speed estimate under switched loading-
2........................................................................................................... 128
Figure 4-69 1500 rpm speed reference, motor speed estimate under switched loading-
1........................................................................................................... 128
Figure 4-70 1500 rpm speed reference, motor speed estimate under switched loading-
2........................................................................................................... 129
Figure 4-71 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-1 .............................................................................. 130
Figure 4-72 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-2 .............................................................................. 131
Figure 4-73 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-1 .............................................................................. 131
Figure 4-74 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-2 .............................................................................. 132
Figure 4-75 500 rpm to -500 rpm speed reference, motor speed estimate under no-
load speed reversal .............................................................................. 133
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Figure 4-76 1000 rpm to -1000 rpm speed reference, motor speed estimate under no-
load speed reversal .............................................................................. 134
Figure 4-77 50 rpm speed reference, motor speed estimate under switched loading
............................................................................................................. 136
Figure 4-78 50 rpm speed reference, stationary reference frame isds current........... 137
Figure 4-79 50 rpm speed reference, stationary reference frame, isqs current.......... 138
Figure 4-80 100 rpm speed reference, motor speed estimate under switched loading
............................................................................................................. 139
Figure 4-81 100 rpm speed reference, stationary reference frame, isds current........ 139
Figure 4-82 100 rpm speed reference, stationary reference frame, isqs current........ 140
Figure 4-83 150 rpm speed reference, motor speed estimate under switched loading
............................................................................................................. 140
Figure 4-84 150 rpm speed reference, stationary reference frame, isds current........ 141
Figure 4-85 150 rpm speed reference, stationary reference frame, isqs current........ 141
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LIST OF SYMBOLS
SYMBOL
emd Back emf d-axis component
emq Back emf q-axis component
ieds d-axis stator current in synchronous frame
ieqs q-axis stator current in synchronous frame
iss Stator current in stationary frame
isds d-axis stator current in stationary frame
isqs q-axis stator current in stationary frame
iar Phase-a rotor current
ibr Phase-b rotor current
icr Phase-c rotor current
ias Phase-a stator current
ibs Phase-b stator current
ics Phase-c stator current
Lm Magnetizing inductance
Lls Stator leakage inductance
Llr Rotor leakage inductance
Ls Stator self inductance
Lr Rotor self inductance
Kk Kalman gain
Pk Kalman filter error covariance matrix
qmd Reactive power d-axis component
qmq Reactive power q-axis component
Rs Stator resistance
Rr Referred rotor resistance
Tem Electromechanical torque
τr Rotor time-constant
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Vas Phase-a stator voltage
Vbs Phase-b stator voltage
Vcs Phase-c stator voltage
Var Phase-a rotor voltage
Vbr Phase-b rotor voltage
Vcr Phase-c rotor voltage
Vss Stator voltage in stationary frame
Vsds d-axis stator voltage in stationary frame
Vsqs q-axis stator voltage in stationary frame
Veds d-axis stator voltage in synchronous frame
Veqs q-axis stator voltage in synchronous frame
Vdc DC-link voltage
we Angular synchronous speed
wr Angular rotor speed
wsl Angular slip speed
kx Kalman filter a priori state estimate
kx Kalman filter a posteriori state estimate
zk Kalman filter measurement
θe Angle between the synchronous frame and the stationary frame
θd Angle between the synchronous frame and the stationary frame when d-axis
is leading
θq Angle between the synchronous frame and the stationary frame when q-axis
is leading
θψr Rotor flux angle
ψss Stator flux linkage in stationary frame
ψsds d-axis stator flux linkage in stationary frame
ψsqs q-axis stator flux linkage in stationary frame
ψeds d-axis stator flux linkage in synchronous frame
ψeqs q-axis stator flux linkage in synchronous frame
ψas Phase-a stator flux linkage
xx
ψbs Phase-b stator flux linkage
ψcs Phase-c stator flux linkage
ψar Phase-a rotor flux linkage
ψbr Phase-b rotor flux linkage
ψcr Phase-c rotor flux linkage
σ Leakage coefficient
1
CHAPTER 1
1.1 Induction Machine Drives
Due to non-linear and complex mathematical model of induction motor, it requires
more sophisticated control techniques compared to DC motors. The scalar V/f
method is able to provide speed control, but this method cannot provide real-time
control. In other words, the system response is only satisfactory at steady state and
not during transient conditions. Dynamic performance of this type of control
methods was unsatisfactory because of saturation effect and the electrical parameter
variation with temperature. This results in excessive current and over-heating, which
necessitates the drive to be oversized. This over-design no longer makes the motor
cost effective due to high cost of the drive circuitry [1].
Recent improvements with reduced loss and fast switching semiconductor power
switches on power electronics, fast and powerful digital signal processors on
controller technology have made advanced control techniques of induction machine
drives feasible and applicable. Thanks to field-oriented control (FOC) schemes [2]-
[3] induction motors can be made to operate with properties similar to those of a
separately excited DC motors.
1.2 The Field Oriented Control (Vector Control) of Induction Machines
Basically, field oriented control (FOC) is a method based on vector coordinates.
The term “vector” refers to the control technique that controls both the amplitude and
the phase of AC excitation voltage. Vector control is used for controllers that
1 INTRODUCTION
2
maintain 90° spatial orientation between the two field components which are d and q
co-ordinates of a time invariant system.
In a field oriented induction motor drive, the field flux and armature mmf are
separately created and controlled based on the vector coordinate transformations.
These projections lead to a structure similar to that of a DC machine control.
The field oriented control is used in most of the induction motor drive applications
in order to obtain high control performance, but it needs motor flux position (rotor
flux angle) information and utilizes AC excitation voltages for the current regulation.
Current regulation is provided with advanced feedback control methods based on the
current measurements taken at the output of excitation voltages supplied from
voltage source inverter (VSI). The rotor flux angle can be measured by using shaft
sensor and that information is utilized by field orientation scheme. However, as
discussed in the current study, sensorless control algorithms eliminate the need for a
shaft sensor.
The induction machine drives without the speed sensor are attractive due to low
cost and high reliability. Therefore, flux and speed estimations have become
particular issues of the field oriented control in the recent years. The main
advantages of speed sensorless induction motor drives are lower cost, reduced size of
the drive machine, elimination of sensor cable and increased reliability.
As it is stated, for implementing vector control, the determination of the rotor flux
position is required. Two basic approaches to determine the rotor flux position angle
have evolved. One of them is the direct field orientation which depends on direct
measurement or estimation of rotor flux magnitude and angle. From the feasibility
point of view, implementation of the direct method is difficult. The other one is the
indirect field orientation which makes use of slip relation in computing the angle of
the rotor flux relative to rotor axis.
1.3 Induction Machine Flux Observation
The rotor flux position could be estimated from the terminal quantities (stator
voltages and currents). This technique requires the knowledge of the stator resistance
3
along with the stator-leakage, and rotor-leakage inductances and the magnetizing
inductance.
The flux observation through direct integration of stator voltage is called Voltage
Model Flux Observer (VMFO) which utilizes the measured stator voltage and
current. Direct integration brings about errors due to the stator resistance voltage
drop and integrator bias. Since the voltage drop on stator resistor at high speeds is
less significant compared to stator voltage drop at lower speeds, the error at low
speeds dominates. In addition, the leakage inductance can significantly affect the
system performance in terms of stability and dynamic response.
sdsi
sqsi
rψθ
Figure 1-1 Inputs and outputs of the voltage model flux observer (VMFO)
Current Model Flux Observer (CMFO) is introduced as an alternative approach in
order to overcome the problems caused by the changes in leakage inductance and
stator resistance at low speed. Current model based observers use the measured stator
currents and rotor velocity. The velocity dependency of the current model is a
drawback since this means that even though using the estimated flux eliminates the
flux sensor, position sensor is still required.
4
sdsisqsi
rw
rψθ
Figure 1-2 Inputs and outputs of the current model flux observer
Several methods are suggested which provide a smooth transition between current
and voltage flux observer models. They combine two stator flux models via a first
order lag-summing network [4]. The smooth transition between current and voltage
models is governed by the rotor flux regulator which makes use of CMFO at low
speeds and VMFO at high speed.
The observer structures VMFO and CMFO are open-loop schemes, based on the
induction machine model and they do not use any feedback for correcting outputs.
Therefore, they are quite sensitive to parameter variations.
Flux estimation through closed-loop state observers is also possible. The
robustness against parameter mismatches and signal noise can be improved by
employing closed-loop observers for the estimation of state variables. State observer
is dependent on induction machine model and machine parameters. Basically, the
observed states are rotor flux, stator currents and rotor speed. Full state observers
could be utilized by using adaptive estimation techniques which makes estimation
accuracy improved.
Speed adaptive flux observer is a closed-loop flux observer which is introduced by
Kubota [5]. Adding an error compensator to the model establishes the closed-loop
observer. The error between induction motor model current and measured current is
used to generate corrective inputs to dynamic subsystem of the stator and the rotor.
5
The rotor speed is also required for adaptive observer; the rotor speed is obtained
through a PI controller, primarily from the current error.
)/(tan 1 sqr
sdr ψψ−
rψθ
sdsψsqsψ
rw
sdsi
sqsi
sdsVs
qsV SpeedAdaptive
FluxObserver
sdsis
qsi
Figure 1-3 Inputs and outputs of the speed adaptive flux observer
Closed-loop Kalman filtering techniques can be based on the complete machine
model. The rotor speed is considered as a state variable and induction motor model
becomes non-linear, so the extended Kalman filter must be applied. The corrective
inputs to the dynamic subsystems of the stator, rotor and mechanical model are
derived so that the error function is minimized. The error function is evaluated on the
basis of the predicted state variables, taking the noise in the measured signals and
parameter deviations into account. The statistical approach reduces the error
sensitivity of the observer.
1.4 Sensorless Vector Control of Induction Machine
To implement vector control, determination of the rotor flux position is required.
Rotor speed or position could be measured by a shaft sensor. Moreover, rotor flux
position could be taken by sensing the air-gap flux with the flux sensing coils.
6
The main drawbacks of using speed/position sensor are high cost, lower system
reliability and special attention to noise. Such problems make sensorless drives
popular. The recent trend in field-oriented control is towards avoiding the use of
speed sensors and using algorithms based on the terminal quantities of the machine
for the estimation of the fluxes. Different solutions for sensorless drives have been
proposed in the past few years.
Saliency based fundamental or high frequency signal injection is one of the flux
and speed estimation techniques. A method involving modulation of the rotor slots
[6] results in a salient rotor, and the saliency can be tracked by imposing a balanced,
three-phase, high-frequency set of harmonics from the inverter. An alternative
method is to use saliency caused by magnetic saturation [7]. A closely related
method is presented in [8]. The main benefit of the methods in [6], [7] and [8] is that
the absolute rotor position can be detected. The advantage of the saliency technique
is that the saliency is not sensitive to actual motor parameters. The methods in [6],
[7] and [8] work also at zero rotor speed. However, extra hardware is required and
high frequency signal injection may cause torque ripples, vibration and audible noise
[9].
The rotor speed can be estimated through nonlinear observers, e.g. [10]-[18].
Alternatively, the rotor speed can be considered as a parameter and estimated using
recursive identification, e.g. [19], [20] and [5]. The latter method can also be
augmented to include machine parameter estimation (inductances, resistances, and
time constants). These methods do not need to rely on harmonics or saliency, and the
hardware requirements are the same as for the digital implementation of vector
control, given that the estimation algorithm is not too complex. Their drawback is
that the rotor speed estimate will be inaccurate if the non-estimated machine
parameters are not known.
7
1.5 Structure of the Chapters
Chapter 2 includes mathematical model of induction machine in terms of reference
frames notation. Field oriented control (FOC), space vector pulse width modulation
technique are also introduced at chapter 2.
Chapter 3 covers observers for sensorless field oriented control of induction motor.
Chapter 4 includes implementation of techniques regarding magnetic flux
estimation and rotor speed estimation by the use of sensorless closed loop observers.
Such that adaptive magnetic flux estimators –a kind of sensorless closed loop
estimation technique- and Kalman filters.
Chapter 5 concludes the overall thesis work of the closed speed loop vector
controlled induction motor.
8
CHAPTER 2
2.1 System Equations in the Stationary a,b,c Reference Frame
The induction machine has two electrically active elements: a rotor and a stator
shown in Figure 2-1. In normal operation, the stator is excited by alternating voltage.
The stator excitation creates a magnetic field in the form of a rotating, or traveling
wave, which induces currents in the circuits of the rotor. Those currents, in turn,
interact with the traveling wave to produce torque. To start the analysis of induction
machine, assume that both the rotor and the stator can be described by the balanced
three phase windings. The two sets are, of course, coupled by mutual inductances
which are dependent on rotor position.
It is assumed that the winding configuration is as in the Figure 2-2. Stator windings
are indicated as as, bs and cs. The as, bs and cs are supposed to have the same number
of effective turns, Ns. The bs and cs are symmetrically displaced from the as by ±120o.
The subscript ‘s’ is used to denote that these windings are stator or stationary
windings. The rotor windings are similarly arranged but have Nr turns. These
windings are designated by ar, br and cr in which second subscript reminds us that
these three windings are rotor or rotating windings. [21]
2 INDUCTION MACHINE MODELING, FIELD ORIENTED
CONTROL and PWM with SPACE VECTOR THEORY
9
Figure 2-1 Axial view of an induction machine
Figure 2-2 Magnetic axes of three phase induction machine
10
The voltage equations (2-1) - (2-8) describing the stator and rotor circuits are well
known and widely referred equations in the literature [21]. Phase voltage equations
can be represented in matrix form.
dtd
irv
dtd
irv
abcrabcrrabcr
abcsabcssabcs
ψ
ψ
+=
+= (2-1)
vabcs, iabcs and ψabcs are 3x1 column vectors defined by
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
cs
bs
as
abcs
cs
bs
as
abcs
cs
bs
as
abcs
iii
ivvv
vψψψ
ψ ; ; (2-2)
Similar definitions apply for the rotor variables vabcr, iabcr and ψabcr.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
cr
br
ar
abcs
cr
br
ar
abcr
cr
br
ar
abcriii
ivvv
vψψψ
ψ ; ; (2-3)
Coupling between stator and rotor phases are given in matrix forms as follows. The
flux linkages are, therefore, related to the machine currents.
)()(
)()(
rabcrsabcrabcr
rabcssabcsabcs
ψψψ
ψψψ
+=
+=
(2-4)
where
abcs
csbcsacs
bcsbsabs
acsabsas
sabcs iLLLLLLLLL
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=)(ψ (2-5)
11
abcr
crcsbrcsarcs
crbsbrbsarbs
crasbrasaras
rabcs iLLLLLLLLL
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
,,,
,,,
,,,
)(ψ (2-6)
abcr
crbcracr
bcrbrabr
acrabrar
rabcr iLLLLLLLLL
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=)(ψ (2-7)
abcs
cscrbscrascr
csbrbsbrasbr
csarbsarasar
sabcr iLLLLLLLLL
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
,,,
,,,
,,,
)(ψ (2-8)
Note that as a result of reciprocity, the inductance matrix in (2-7), is simply the
transpose of the inductance matrix of (2-6), because mutual inductances are equal.
(i.e., asbrbras LL ,, = )
2.1.1. Determination of Induction Machine Inductances [21]
The mutual inductance between a winding x and a winding y is determined by:
απμ cos40 ⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
grlNNL yxxy (2-9)
where r is the radius, l is the length of the axial length of stator and g is the length
of airgap. Nx is the number of effective turns of the winding x and Ny is the number
of effective turns of the winding y. Finally, let α be the angle between magnetic axes
of the phases x and y.
The self inductance of stator phase as winding is obtained by simply setting α=0,
and by setting both Nx and Ny in (2-9) to Ns as
⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
42
0πμ
grlNL sam (2-10)
12
The subscript m is used to denote the fact that this inductance is magnetizing
inductance. That is, it is associated with flux lines which cross the air gap and link
rotor as well as stator windings. In general, it is necessary to add a relatively smaller,
but more important leakage term to (2-10) to account for leakage flux. This term
accounts for flux lines which do not cross the gap but instead close to the stator slot
itself (slot leakage) in the air gap (belt and harmonic leakage) and at the ends of the
machine (end winding leakage). Hence, the total self inductance of phase as can be
expressed.
amlsas LLL += (2-11) where Lls represents the leakage term. Since the windings of the bs and the cs phases
are identical to phase as, it is clear that the magnetizing inductances of these
windings are the same as phase as so that
cmlscs
bmlsbs
LLL
LLL
+=
+= (2-12)
It is apparent that Lam, Lbm, Lcm are equal making the self inductances also equal. It
is, therefore, useful to define stator magnetizing inductance
⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
42
0πμ
grlNL sms (2-13)
so that
mslscsbsas LLLLL +=== (2-14)
The mutual inductance between phases as and bs, bs and cs, and cs and as is
derived by simply setting α=2π/3 and Nx =Ny=Ns in (2-9). The result is
13
⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−===
82
0πμ
grlNLLL scasbcsabs (2-15)
or, in terms of (2-13),
2ms
casbcsabsLLLL −=== (2-16)
The flux linkages of phases as, bs and cs resulting from currents flowing in the
stator windings can be now expressed in matrix form as
abcs
mslsmsms
msmsls
ms
msmsmsls
sabcs i
LLLL
LLLL
LLLL
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+−−
−+−
−−+
=
22
22
22
)(ψ (2-17)
Let us now turn our attention to the mutual coupling between the stator and rotor
windings. Referring to Figure 2-2, we can see that the rotor phase ar is displaced by
stator phase as by the electrical angle θr where θr in this case is a variable. Similarly,
the rotor phases br and cr are displaced from stator phases bs and cs by θr respectively.
Hence, the corresponding mutual inductances can be obtained by setting Nx=Ns,
Ny=Nr, and α= θr in (2-9).
rmss
r
rrscrcsbrbsaras
LNN
grlNNLLL
θ
θπμ
cos
cos40,,,
=
⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛===
(2-18)
The angle between the as and br phases is θr+2π/3, so that
( )3/2cos,,, πθ +=== rmss
rarcscrbsbras L
NNLLL (2-19)
14
Finally, the stator phase as is displaced from the rotor cr phase by angle 3/2πθ −r .
Therefore,
( )3/2cos,,, πθ −=== rmss
rbrcsarbscras L
NNLLL (2-20)
The above inductances can now be used to establish the flux linking the stator
phases due to currents in the rotor circuits. In matrix form,
( ) ( )
( ) ( )( ) ( )
abcr
rrr
rrr
rrr
mss
rrabcs iL
NN
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−++−−+
=θπθπθπθθπθπθπθθ
ψcos3/2cos3/2cos
3/2coscos3/2cos3/2cos3/2coscos
)( (2-21)
The total flux linking the stator windings is clearly the sum of the contributions
from the stator and the rotor circuits, (2-17) and (2-21),
)()( rabcssabcsabcs ψψψ += (2-22)
It is not difficult to continue the process to determine the rotor flux linkages. In
terms of previously defined quantities, the flux linking the rotor circuit due to rotor
currents is
abcr
mss
rlrms
s
rms
s
r
mss
rms
s
rlrms
s
r
mss
rms
s
rms
s
rlr
rabcr i
LNNLL
NNL
NN
LNNL
NNLL
NN
LNNL
NNL
NNL
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+
=
222
222
222
)(
21
21
21
21
21
21
ψ (2-23)
where Llr is the rotor leakage inductance. The flux linking the rotor windings due to
currents in the stator circuit is
15
( ) ( )( ) ( )( ) ( )
abcs
rrr
rrr
rrr
mss
rsabcr iL
NN
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
+−−++−
=θπθπθπθθπθπθπθθ
ψcos3/2cos3/2cos
3/2coscos3/2cos3/2cos3/2coscos
)( (2-24)
Note that the matrix of (2-24) is the transpose of (2-21).The total flux linkages of
the rotor windings are again the sum of the two components defined by (2-23) and
(2-24), that is
)()( sabcrrabcrabcr ψψψ += (2-25)
2.1.2. Three-Phase to Two-Phase Transformations
The performance of three-phase AC machines is described by their voltage
equations and flux linkages. Some machine inductances are also functions of rotor
position. The coefficients of the differential equations, which describe the behavior
of these machines, are time-varying except when the rotor is stalled. A change of
variables is often used to reduce the complexity of these differential equations. Using
transformations, many properties of electric machines can be studied without
complexities in the voltage equations. These transformations make it possible for
control algorithms to be implemented on the DSP. For this purpose, the method of
symmetrical components uses a complex transformation to decouple the abc phase
variables. By this approach, many of the basic concepts and interpretations of this
general transformation are concisely established.
2.1.2.1. The Clarke Transformation [1]
The transformation of stationary circuits to a stationary reference frame was
developed by E. Clarke [22]. The stationary two-phase variables of Clarke’s
transformation are denoted as α and β. As shown in Figure 2-3, α-axis and β-axis are
orthogonal.
16
Figure 2-3 Relationship between the α, β and the abc quantities [23]
The symbol f is used to represent any of the three phase stator circuit variables
such as voltage, current or flux linkage, variables along a, b and c axes
( cba fandff , ) can be reffered to the stationary two-phase variables α, β and zero
sequence ( 0, fandff βα ) by,
]][[][ 00 abcfTf αβαβ = (2-26)
where
Tffff ][][ 00 βααβ =
Tcbaabc ffff ][][ =
The transformation matrix is defined as
17
[ ]
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
=
21
21
21
23
230
21
211
32
0αβT (2-27)
And inverse transformation matrix is presented by
[ ]
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−=−
123
21
123
21
1011
0αβT (2-28)
2.1.2.2. The Park Transformation [1]
In the late 1920s, R.H. Park [24] introduced a new approach to electric machine
analysis. He formulated a change of variables which replaced variables such as
voltages, currents, and flux linkages associated with fictitious windings rotating with
the rotor. He referred the stator and rotor variables to a reference frame fixed on the
rotor. From the rotor point of view, all the variables can be observed as constant
values. Park’s transformation, a revolution in machine analysis, has the unique
property of eliminating all time varying inductances from the voltage equations of
three-phase ac machines due to the rotor spinning.
Although changes of variables are used in the analysis of AC machines to
eliminate time-varying inductances, changes of variables are also employed in the
analysis of various static and constant parameters in power system components.
Fortunately, all known real transformations for these components are also contained
in the transformation to the arbitrary reference frame. The same general
transformation used for the stator variables of ac machines serves as the rotor
18
variables of induction machines. Park’s transformation is a well-known three-phase
to two-phase transformation in machine analysis.
Park’s transformation presented in Figure 2-4 transforms three-phase quantities fabc
into two-phase quantities developed on a rotating dq0 axes system, whose speed is w.
Figure 2-4 Relationship between the dq and the abc quantities [23]
])][([][ 00 abcdqdq fTf θ= (2-29)
where
Tqddq ffff ][][ 00 =
Tcbaabc ffff ][][ =
where the dq0 transformation matrix is defined as:
19
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ −−−
⎟⎠⎞
⎜⎝⎛ +⎟
⎠⎞
⎜⎝⎛ −
=
21
21
21
32sin
32sinsin
32cos
32coscos
32)]([ 0
πθπθθ
πθπθθ
θdqT (2-30)
and the inverse is given by:
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ +
⎟⎠⎞
⎜⎝⎛ −−⎟
⎠⎞
⎜⎝⎛ −
−
=−
13
2sin3
2cos
13
2sin3
2cos
1sincos
)]([ 10
πθπθ
πθπθ
θθ
θdqT (2-31)
where θ is the angle between the phase a- axis and d - axis. and can be calculated by
∫ +=t
dw0
)0()( θττθ (2-32)
where τ is the dummy variable of integration.
2.2 Reference Frames
2.2.1. Induction Motor Model in the Arbitrary dq0 Reference Frame
The coupling between the stator and rotor circuits can be eliminated if the stator
and the rotor equations are referred to a common frame of reference. The reference
frames are usually selected on the basis of conveniences or computational reduction.
A common frame of reference can be non-rotating (i.e. w = 0) which it is associated
with the stator and it is, therefore, called as the stator or stationary reference frame
20
with a frame notation dsqs. Alternatively, dq0 axes (i.e. the common frame) can be
taken to rotate with the same angular velocity (i.e. w = ws, synchronous speed), as
the rotor circuits, and is termed as the rotor reference frame with a frame notation
deqe. It may even be useful to select these axes synchronously rotating at w with one
of the complex vectors denoting stator or rotor voltage, current or even flux as
arbitrary reference frame. Each reference frame has appealing advantages. For
example, stationary reference frame, the dsqs variables of the machine are in the same
frame as those normally used for the supply network. Furthermore, at the
synchronously rotating frame, the deqe variables are DC in steady state.
Once the equations of the induction machine are derived in the arbitrary reference
frame, which is rotating at a speed w, in the direction of the rotor rotation, the
transformation between reference frames could be obtained easily. When the
induction machine runs in the stationary frame, these equations of the induction
machine can then be achieved by setting w = 0. These equations can also be obtained
in the synchronously rotating frame by setting w = we.
In matrix notation, the stator winding abc voltage equations can be expressed as:
dtd
irv abcsabcssabcs
ψ+= (2-33)
Applying transformation to the stator windings abc voltages, the stator winding qd0
voltages in arbitrary reference frame are obtained.
sdqsdqsdq
sdqsdq irdt
dwv 00
000
000001010
++⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
ψψ (2-34)
where
21
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡==
100010001
0 ssdq rranddtdw θ (2-35)
Likewise, the rotor voltage equation becomes:
rdqrdqrdq
rdqrrdq irdt
dwwv 00
000
000001010
)( ++⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−=
ψψ (2-36)
Stator and rotor flux linkage equations are given as;
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
′′
′
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
′′
′=
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
′′
′
r
dr
qr
s
ds
qs
lr
rm
rm
ls
ms
ms
r
dr
qr
s
ds
qs
ii
iii
i
LLL
LLL
LLLL
0
0
0
0
00000000000000000000000000
ψψ
ψψψ
ψ
(2-37)
where primed quantities denote referred values to the stator side.
mlrr
mlss
LLL
LLL
+′=′
+= (2-38)
And
lrr
slrsmsm L
NN
LgrlNLL 2
02 )(,
423
23
=′⎟⎟⎠
⎞⎜⎜⎝
⎛==
πμ (2-39)
Electromagnetic torque, Tem equation is given as,
[ ] NmiiwwiiwwpT drqrqrdrrdsqsqsds
rem ))(()(
223 ′′−′′−+−= ψψψψ (2-40)
22
Using the flux linkage relationships from (2.37), (2-40) can be simplified as,
NmiiiiLp
Nmiip
NmiipT
dsqrqsdrm
dsqsqsds
qrdrdrqrem
)(22
3
)(22
3
)(22
3
′−′=
−=
′′−′′=
ψψ
ψψ
(2-41)
2.2.2. Induction Motor Model in dq0 Stationary and Synchronous
Reference Frames
Once the induction motor model in the arbitrary dq0 reference frame is established,
dq0 stationary (denoted as dsqs) and synchronous (denoted as deqe) reference frame
equations can be derived. To distinguish these two frames from each other, an
additional superscript will be used, s for stationary frame variables and e for
synchronously rotating frame variables.
i. dq0 stationary frame induction motor equations are given as (2-42) - (2-45).
Stator qsds voltage equations:
dss
sds
s
dss
qss
sqs
s
qss
irdt
dv
irdt
dv
+=
+=
ψ
ψ
(2-42)
Rotor qsds voltage equations:
23
drs
rqrs
rdr
s
drs
qrs
rdrs
rqr
s
qrs
irwdt
dv
irwdt
dv
′′+′+′
=′
′′+′−+′
=′
ψψ
ψψ
)(
)(
(2-43)
where
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
′′
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
′′
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
′′
drs
qrs
dss
qss
rm
rm
ms
ms
drs
qrs
dss
qss
iiii
LLLL
LLLL
0000
0000
ψψψψ
(2-44)
Torque Equations:
NmiiiiLp
Nmiip
NmiipT
dss
qrs
qss
drs
m
dss
qss
qss
dss
qrs
drs
drs
qrs
em
)(22
3
)(22
3
)(22
3
′−′=
−=
′′−′′=
ψψ
ψψ
(2-45)
ii. dq0 synchronous frame induction motor equations are given as (2-46) - (2-49).
Stator qede voltage equations:
24
dse
sqse
eds
e
dse
qse
sdse
eqs
e
qse
irwdt
dv
irwdt
dv
+−=
++=
ψψ
ψψ
(2-46)
Rotor qede voltage equations:
dre
rqre
redr
e
dre
qre
rdre
reqr
e
qre
irwwdt
dv
irwwdt
dv
′′+′−−′
=′
′′+′−+′
=′
ψψ
ψψ
)(
)(
(2-47)
where
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
′′
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
′′
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
′′
dre
qre
dse
qse
rm
rm
ms
ms
dre
qre
dse
qse
iiii
LLLL
LLLL
0000
0000
ψψψψ
(2-48)
Torque Equations:
Nmiip
Nmiip
T
dse
qse
qse
dse
qre
dre
dre
qre
em
)(22
3
)(22
3
ψψ
ψψ
−=
′′−′′=
(2-49)
25
2.3 Field Oriented Control (FOC)
Following the concepts outlined for the DC machine, the requirements are for
torque and flux control which has to be also satisfied for ac machines in order to
implement successful field orientation control [21]. They can be basically stated as:
• Independent control of the armature current to overcome the effects of armature
winding resistance, leakage inductance and induced voltage.
• Independent control of flux at a constant value.
• Independent control of orthogonality between the flux and magnetomotive force
(MMF) axes to avoid interaction of MMF and flux.
If all of these three requirements are met at all times, the torque will follow the
current, which will allow an instantaneous torque control and decoupled flux and
torque regulation.
In the DC machine, first and second requirements are assured by the presence of
the commutator and the separate field excitation system. In AC machines, these two
requirements are achieved by external controls.
Next, a two phase dq model of an induction machine rotating at the synchronous
speed is introduced which will help to carry out this decoupled control concept to the
induction machine. This model can be summarized by the following equations:
dse
sqse
eds
e
dse irw
dtdv +−= ψψ (2-50)
qse
sdse
e
eqs
qse irw
dtd
v ++= ψψ
(2-51)
( ) qre
rdre
re
eqr irww
dtd
+−+= ψψ
0 (2-52)
( ) dre
rqre
re
edr irww
dtd
+−−= ψψ
0 (2-53)
26
qre
meqss
eqs iLiL ′+=ψ (2-54)
dre
medss
eds iLiL ′+=ψ (2-55)
qre
reqsm
eqr iLiL ′′+=′ψ (2-56)
dre
redsm
edr iLiL ′′+=′ψ (2-57)
( )eds
eqr
eqs
edr
r
mem ii
LLpT ψψ ′−′=
23 (2-58)
Lrr
em TBwdt
dwJT ++= (2-59)
In this model, it can be seen from the torque expression (2-58) that if the rotor flux
along the q-axis is zero, then all the flux is aligned along the d-axis and therefore, the
torque can be instantaneously controlled by controlling the current along q-axis. The
qe-axis is set perpendicular to the de-axis. The flux along the qe-axis in that case will
obviously be zero. The phasor diagram Figure 2-7 shows these axes. The angle eθ
keeps changing as the machine input currents change. The angle eθ accurately
known, d-axis of the deqe frame can be locked to the flux vector. The Park and the
Clarke transformations at the machine side is represented in Figure 2-5.
27
Figure 2-5 The Park and the Clarke transformations at the machine side [1]
The control inputs at field oriented control can be specified in terms of two-phase
synchronous frame ieds and ie
qs variables. ieds is aligned along the de-axis i.e. the flux
vector, so does ieqs with the qe-axis. These two-phase synchronous control inputs are
first converted into two-phase stationary ones and then to three-phase stationary
control inputs. This can be achieved by taking the inverse transformation of variables
from the arbitrary rotating reference frame to the stationary reference frame and then
to the abc system. To accomplish this, the flux angle eθ must be known precisely.
The block diagram of this procedure is shown in Figure 2-6. In this block diagram, *
is a representation of commanded or desired values of variables.
The angle eθ can be found either by Indirect Field Oriented Control (IFOC) or by
Direct Field Oriented Control (DFOC). The controller implemented in this fashion
that can achieve a decoupled control of the flux and the torque is known as field
oriented controller. The block diagram is as in Figure 2-10.
28
Figure 2-6 Variable transformation in the field oriented control [1]
The absence of the field angle sensors, along with the ease of operation at low
speeds, has increased the popularity of the indirect vector control strategy. While the
direct method is inherently the most desirable scheme, it suffers from the
unreliability in measuring the flux. Although the indirect method can approach the
performance of the direct measurement scheme, its major weakness is the accuracy
of the control gain, which heavily depends on the motor parameters. The block
diagrams of indirect field oriented control and direct field oriented control are
illustrated at Figure 2-8 and Figure 2-9 respectively.
29
Figure 2-7 Phasor diagram of the field oriented drive system
Figure 2-8 Indirect field oriented drive system
As it can be seen from Figure 2-8, indirect field orientation drive system needs the
rotor resistance or rotor time-constant as a parameter. Accurate knowledge of the
rotor resistance is essential to achieve the highest possible efficiency from the control
structure. Lack of this knowledge results in detuning of the FOC.
30
Figure 2-9 Direct field oriented drive system
Figure 2-10 shows the block diagram of indirect field orientation control strategy
with sensor in which speed regulation is possible using a control loop.
Figure 2-10 Indirect field oriented induction motor drive system with sensor [1]
31
As shown in Figure 2-10, two-phase current feeds the Clarke transformation block.
These projection outputs are indicated as isds and is
qs. These two components of the
current provide the inputs to Park’s transformation, which gives the currents in qdse
the excitation reference frame. The ieds and ieqs components, which are outputs of the
Park transformation block, are compared to their reference values ie*ds, the flux
reference, and ie*qs, the torque reference. The torque command, ie*
qs, comes from the
output of the speed controller. The flux command, ie*ds , is the output of the flux
controller which indicates the right rotor flux command for every speed reference.
Magnetizing current ie*ds is usually between 40 and 60% of the nominal current [2].
For operating in speeds above the nominal speed, a field weakening section should
be used in the flux controller section. The current regulator outputs, ve*ds and ve*
qs are
applied to the inverse Park transformation. The outputs of this projection are vsds and
vsqs, which are the components of the stator voltage vector in dsqs the orthogonal
reference frame. They form the inputs of the SVPWM block. The outputs of this
block are the signals that drive the inverter.
2.4 Space Vector Pulse Width Modulation (SVPWM)
2.4.1. Voltage Fed Inverter (VSI)
The voltage source inverters (VSI) are the most common power electronics
converters. The block diagram of the voltage source inverter supplied form the
uncontrolled rectifier is shown in Figure 2-11. The DC link capacitor constitutes the
actual voltage source, since voltage across it cannot change instantly. Since the
output voltage of the diode bridge rectifier is not a pure DC, a filter inductor is
included to absorb ripple component.
32
Figure 2-11 The block diagram of VSI supplied from a diode rectifier
A diagram of a three phase VSI is shown in the Figure 2-12.
Figure 2-12 Three phase voltage source inverter supplying induction motor [1]
As it can be seen from Figure 2-11 and Figure 2-12, voltage source inverter has
bridge topology with three branches (phases), each consisting of two power switches
33
and two freewheeling diodes. The inverter here is supplied from an uncontrolled,
diode-based rectifier, via DC link which contains an LC filter in the inverted
configuration. The uncontrolled rectifier allows the power flow from the supply to
the load only.
2.4.2. Voltage Space Vectors
In terms of the desired phase voltages, the voltage space vector can be written by
multiplying phase voltages by their phase orientations.
3/4_
3/2_
0_
)( ππ jcn
jbn
jans eVeVeVtV ⋅+⋅+⋅= (2-60)
A switch in a VSI is either “up” or “down”, with the instantaneous output voltage
either 1 or 0 times of Vdc . With three branch, eight switch-status combinations are
possible.The voltage space vector can instantly take on one of the following seven
distinct instantaneous values as shown in Table 2-1.
Table 2-1 Instantaneous Basic Voltage Vectors [25]
Switching State
S5 S3 S1
Basic Vector Value
0 0 0 )000(0
−
v 0
0 0 1 )001(1
−
v 0j
dc eV ⋅
0 1 0 )010(2
−
v 3/2πj
dc eV ⋅
0 1 1 )011(3
−
v 3/πj
dc eV ⋅
34
Table 2-1 (Cont’d)
1 0 0 )100(4
−
v 3/4πj
dc eV ⋅
1 0 1 )101(5
−
v 3/5πj
dc eV ⋅
1 1 )110(6
−
v πj
dc eV ⋅
1 1 1 )111(7
−
v 0
In Table 2-1, −
1v and −
7v are the zero vectors. The resulting instantaneous voltage
vectors, which are called the “basic vectors”, are shown in Figure 2-13. The basic
vectors form six sectors in Figure 2-13.
Figure 2-13 Basic space vectors [25]
35
2.4.3. SVPWM Application to the Static Power Bridge
Space Vector PWM (SVPWM) refers to a special technique of determining the
switching sequence of the upper three power transistors of a three-phase voltage
source inverter (VSI). It has been shown to generate less harmonic distortion in the
output voltages or current in the windings of the motor. SVPWM provides more
efficient use of the DC bus voltage compared to the direct sinusoidal modulation
technique.
In AC drive applications, voltage sources are not sinusoidal. Instead, they are
replaced by 6 power switches which act as on/off to the rectified DC bus voltage.
The aim is to create sinusoidal current in the windings to generate rotating field.
Owing to the inductive nature of the phases, a pseudo sinusoidal current is created by
modulating the duty-cycle of the power switches. The switches shown in the Figure
2-12 are activated by signals a, b, c and their complement values. Eight different
combinations are available with this three phase voltage source inverter including
two zero states. It is possible to express each phase to neutral voltages in terms of DC
supply voltage Vdc, for each switching combination of switches as listed in Table
2-2.
The voltages, Van, Vbn, and Vcn are the output voltages applied to the windings of
a motor. The six power transistors which are controlled by a, a’, b, b’, c and c’ gating
signals and shape the output voltages. When an upper transistor is switched on, i.e.,
when a, b, and c are 1, the corresponding lower transistor is switched off, i.e., the
corresponding a’, b’ or c’ is 0. The on and off states of the upper transistors S1, S3,
and S5, or the states of a, b, and c are sufficient to evaluate the output voltage.
Table 2-2 Power bridge output voltages (Van, Vbn, Vcn)
Switch Positions Phase Voltages
S5 S3 S1 Van Vbn Vcn
0 0 0 0 0 0
36
Table 2-2 (Cont’d)
0 0 1 2Vdc/3 -Vdc/3 -Vdc/3
0 1 0 -Vdc/3 2Vdc/3 -Vdc/3
0 1 1 Vdc/3 Vdc/3 -2Vdc/3
1 0 0 -Vdc/3 -Vdc/3 2Vdc/3
1 0 1 Vdc/3 -2Vdc/3 Vdc/3
1 1 0 -2Vdc/3 Vdc/3 Vdc/3
1 1 1 0 0 0
Since only 8 combinations are possible for the power switches, Vs
ds, Vsqs can also
take finite number of values in the (dsqs) frame. According to the command signals a,
b, c Table 2-3 includes stator voltages in (dsqs) frame.
Table 2-3 Stator voltages in (dsqs) frame and related voltage vector
Switch Positions (dsqs) frame Voltages
S5 S3 S1 Vsds Vs
qs Vectors
0 0 0 0 0 )000(0
−
v
0 0 1 2Vdc/3 0 )001(1
−
v
0 1 0 -Vdc/3 Vdc/√3 )010(2
−
v
0 1 1 Vdc/3 Vdc/√3 )011(3
−
v
1 0 0 -Vdc/3 -Vdc/√3 )100(4
−
v
1 0 1 Vdc/3 -Vdc/√3 )101(5
−
v
37
Table 2-3 (Cont’d)
1 1 0 -2Vdc/3 0 )110(6
−
v
1 1 1 0 0 )111(7
−
v
The eight voltage vectors re-defined by the combination of the switches are
represented in Figure 2-14.
Given a reference voltage (derived from the inverse Park transform), the following
step is used to approximate this reference voltage by the above defined eight vectors.
The method used to approximate the desired stator reference voltage with only eight
possible states of switches combines adjacent vectors of the reference voltage and
modulates the time of application of each adjacent vector. In Figure 2-14 for a
reference voltage Vsref is in the third sector and the application time of each adjacent
vector is given by:
33
11
031
VTTV
TTV
TTTT
sref +=
++= (2-61)
where T1 , T3, and T0 are respective time shares for vectors V1 and V3 an null vector
V0 within period T.
38
Figure 2-14 Projection of the reference voltage vector
The determination of the amount of times T1 and T3 are given by simple
projections:
)60(
)30cos(
0
11
03
3
tgV
x
xVTTV
VTTV
ssqref
ssdref
ssqref
=
+=
=
(2-62)
Finally, with the (dsqs) component values of the vectors given in Table 2-3, the
duration periods of application of each adjacent vector are:
39
)3(21
ssq
ssd VVTT −= (2-63)
s
sqTVT =3 (2-64)
where the vector magnitudes are “ 3/2 dcV ” and both sides are normalized by
maximum phase to neutral voltage 3/DCV . The rest of the period spent in applying the null vector (T0=T-T1-T3). For every
sector, commutation duration is calculated. The amount of times of vector
application can all be related to the following variables:
ssd
ssq
ssd
ssq
ssq
VVZ
VVY
VX
23
21
23
21
−=
+=
=
(2-65)
In the previous example for sector 3, T1 = -TZ and T3 =TX. Extending this logic,
one can easily calculate the sector number belonging to the related reference voltage
vector. Then, three phase quantities are calculated by inverse Clarke transform to get
sector information. The following basic algorithm helps to determine the sector
number systematically.
If ssqref VV =1 > 0 then set A=1 else A=0
If )3(21
2s
sqs
sdref VVV −= > 0 then set B=1 else B=0
If )3(21
3s
sqs
sdref VVV −−= > 0 then set C=1 else C=0
Then,
40
Sector = A+2B+4C
The duration of the sector boundary vectors application after normalizing with the
period T can be determined as in Table 2-4.
Table 2-4 Durations of sector boundary
Sector t1 t2
1: t1= Z t2= Y
2: t1= Y t2=-X
3: t1=-Z t2= X
4: t1=-X t2= Z
5: t1= X t2=-Y
6: t1=-Y t2=-Z
Saturations should be applied to the durations of t1 and t2 in case following
saturation condition is satisfied.
If (t1+ t2) > PWM period then;
t1sat = t1/( t1+t2)*PWM period and t2sat =t2/( t1+t2)*PWM period
The third step is to compute the three necessary duty-cycles. This is shown below:
41
2
1
21
2
ttt
ttt
ttperiodPWMt
boncon
aonbon
aon
+=
+=
−−=
(2-66)
The last step is to assign the right duty-cycle (txon) to the right motor phase (in other
words, to the Ta, Tb and Tc) according to the sector. Table 2-5 depicts this
determination below (i.e., the on time of the inverter switches).
Table 2-5 Assigned duty cycles to the PWM outputs
1 2 3 4 5 6
Ta tbon taon taon tcon tbon tcon
Tb taon tcon tbon tbon tcon taon
Tc tcon tbon tcon taon taon tbon
The phase voltage of a general 3-phase motor Van, Vbn, Vcn can be calculated
from the DC-bus voltage (Vdc), and three upper switching functions of inverter S1,
S3, and S5. The 3-ph windings of motor are connected either Δ or Y without a neutral
return path (or 3-ph, 3-wire system).
Each phase of the motor is simply modeled as a series impedance of resistance r
and inductance L and back emf ea, eb, ec. Thus, three phase voltages can be computed
as:
42
aa
anaan edtdiLriVVV ++=−= (2-67)
bb
bnbbn edtdiLriVVV ++=−= (2-68)
cc
cnccn edtdiLriVVV ++=−= (2-69)
Summing these three phase voltages yields
cbacba
cbancba eeedt
iiidLriiiVVVV +++++
+++=−++)()(3 (2-70)
For a 3-phase system with no neutral path and balanced back emfs, ia+ib+ic=0, and
ea+eb+ec,=0. Therefore, (2-71) becomes Van+Vbn+Vcn,=0. Furthermore, the neutral
voltage can be simply derived from (2-71) as
)(31
cban VVVV ++= (2-71)
Now three phase voltages can be calculated as:
cbacbaaan VVVVVVVV31
31
32)(
31
−−=++−= (2-72)
cabcbabbn VVVVVVVV31
31
32)(
31
−−=++−= (2-73)
43
baccbaccn VVVVVVVV31
31
32)(
31
−−=++−= (2-74)
Three voltages Va, Vb, Vc are related to the DC-bus voltage Vdc and three upper
switching functions S1, S3, and S5 as:
dca VSV 1= (2-75)
dcb VSV 3= (2-76)
dcc VSV 5= (2-77)
where S1, S3, and S5 =either 0 or 1, and S2=1-S1, S4=1-S3, and S6=1-S5.
As a result, three phase voltages in (2-82) to (2-84) can also be expressed in terms
of DC-bus voltage and three upper switching functions as:
)31
31
32( 531 SSSVV dcan −−= (2-78)
)31
31
32( 513 SSSVV dcbn −−=
(2-79)
)31
31
32( 315 SSSVV dccn −−= (2-80)
It is emphasized that the S1, S3, S5 are defined as the upper switching functions. If
the lower switching functions are available instead, then the out-of-phase correction
44
of switching function is required in order to get the upper switching functions as
easily computed from equation (S2=1-S1, S4=1-S3, and S6=1-S5). Next the Clarke
transformation is used to convert the three phase voltages Van, Vbn, and Vcn to the
stationary dq-axis phase voltages Vsds and Vs
qs. Because of the balanced system (Van
+ Vbn + Vcn=0) Vcn is not used in Clarke transformation.
45
CHAPTER 3
In general an estimator is defined as a dynamic system whose state variables are
estimates of some other systems (e.g. induction motor). There are basically two
forms of estimators; open-loop estimator and closed-loop estimator. The difference
between them is whether a correction term is used to adjust the response of the
estimator or not. A closed-loop estimator is referred as an observer.
Various open-loop flux estimators (flux models) are investigated in the literature.
Different types of flux estimators could be implemented in rotor-flux-oriented
reference frame or stationary reference frame. The common input to these models is
monitored rotor speed or monitored rotor position. For a speed sensorless drive
system, it is not possible to use such models. However, it is possible to establish a
flux model which uses the monitored values of the stator voltages and stator currents.
It is called improved flux model.
Open-loop flux estimators using pure integration are sensitive to parameter
variations, and measurement error. These effects become more important at low
stator frequencies due to the dominant effect of stator ohmic drops. An accurate
compensation against ohmic voltage drops must be made prior to the integration.
Due to the temperature dependency of the stator resistance value, it is difficult to
have such compensation. Yet, with such an implementation, a lower frequency limit
for useful operation is approximately 3 Hz with a 50 Hz supply [26].
Various open-loop rotor speed and rotor slip-frequency estimators are obtained by
considering the voltage equations of the induction motor. They generally utilize the
estimates of stator or rotor flux linkages. Hence, open-loop rotor speed estimators
3 OBSERVERS FOR SENSORLESS FIELD ORIENTED
CONTROL OF INDUCTION MACHINE
46
basically rely on open-loop flux estimators. Open-loop rotor speed estimators mainly
use the monitored stator voltages and currents. Some of these open-loop rotor speed
estimation schemes are used in commercial speed sensorless induction motor drives.
However, it is important to note that in general, the accuracy of open-loop estimators
depends on machine parameters used. At low rotor speed, the accuracy of open-loop
estimator is reduced, and in particular, parameter deviations from their actual values
have great influence on the steady-state and transient performance of the drive
system which uses an open-loop estimator.
On-line identification techniques are used in order to reduce the influence of
parameter variations. Stator resistance, rotor time-constant, stator transient
inductance and stator self inductance are identified during a self-commissioning
stage of a vector controlled induction motor drive. It should be noted that in this self
commissioning stage, the induction motor is at standstill during realization of all
measurements. Moreover, adaptive on-line identification techniques are considered
to track parameter variations during operation.
Closed-loop estimators (observers) can be classified according to the type of
representation used for the plant to be observed. Once the plant is considered as
deterministic, the observer is deterministic; or else, the observer is stochastic.
Thanks to the correction term, closed-loop estimators have generally better
performance with respect to open-loop types. For that reason, closed-loop estimators
are investigated throughout this study.
Various types of observers could be used in high-performance induction motor
drives. Full-order state-observer (speed adaptive flux observer) and Kalman filter
types are implemented in this thesis work, where full-order state observer is
deterministic whereas Kalman filter types are stochastic.
This chapter focuses on both speed and flux observers.
47
3.1. Speed Adaptive Flux Observer for Induction Motor
In an inverter-fed electrical drive system, a speed adaptive flux observer could be
used to estimate rotor flux linkage and stator current components by monitoring
stator currents and the monitored (or reconstructed) stator voltages.
A state observer is a closed-loop estimator which can be used for the state (and/or
parameter) estimation of a non-linear dynamic system in real time. In the
calculations, the states are predicted by using a mathematical model of the observed
system (the estimated states and actual states being denoted by x and x
respectively), but the predicted states are continuously corrected by adding a
feedback correction scheme. This correction scheme contains a weighted difference
of some of the measured and estimated output signals (The difference is multiplied
by the observer feedback gain, G). Based on the deviation from the estimated value,
the state observer provides an optimum estimated output value ( x ) at the next input
instant. In an induction motor drive, a state observer can also be used for the real-
time estimation of the rotor speed and some of the machine parameters such as stator
resistance.
3.1.1. Flux Estimation Based on the Induction Motor Model
The model of the induction machine is established in the stationary reference frame
as;
[ ] BuAxxdtd
+= (3-1)
where
[ ]T xsr
ssix 14ψ= ,
442221
1211
xAAAA
A ⎥⎦
⎤⎢⎣
⎡= , [ ] 12x
ssVu= , and [ ]TxBB 241 0=
48
Then, the (3-1) takes the form below,
ss
sss
rsr
BVAx
VB
AAAA
dtd
+=
⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡0
ii 1ss
2221
1211ss
ψψ (3-2)
and
xC ⋅=ssi (3-3)
xDsr ⋅=ψ (3-4)
where
[ ]Tsqs
sdss iii =
[ ]Tsqr
sdr
sr ψψψ =
[ ]Tsqs
sds
ss VVV =
( ) ( ) ( ) IaILRA rrSs 1111 /1/ =−+−= στσσ
( ) ( ) JaIaJILLLA irrrrsm 121212 /1/ +=−= ωτσ
( ) IaILA rrm 2121 / == τ
( ) JaIaJIA irrr 222222 /1 +=+−= ωτ
49
( ) IbILB s 11 /1 == σ
⎥⎦
⎤⎢⎣
⎡=
00100001
C
⎥⎦
⎤⎢⎣
⎡=
10000100
D
⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡=
1001
1001
JI
( )rs
m
LLL 2
1−=σ
r
rr R
L=τ
By using the mathematical model of the induction motor given in (3-1) and adding
a correction term, which contains the weighted difference of actual and the estimated
states, a full-order state observer which estimates the stator current and the rotor flux
linkages can be described as follows,
( )ss
ss
ss iiGBVxAx
dtd ˆˆˆˆ −++= (3-5)
where
⎥⎦
⎤⎢⎣
⎡=
2221
1211
ˆˆˆAAAAA (3-6)
50
xCi ss ˆˆ = (3-7)
xDsr ˆˆ =ψ (3-8)
( ) ( ) JaIaJILLLA irrrrsm 121212 ˆˆ/1/ˆ +=−= ωτσ
( ) JaIaJIA irrr 222222 ˆˆ/1ˆ +=+−= ωτ
where x denotes the estimated values and G is the observer gain matrix which is
selected so that the observer can be stable.
It can be seen from (3-6), A is a function of estimated rotor speed rω .
3.1.1.1. Estimation of Rotor Flux Angle
Once the equations (3-5) - (3-8) are solved, the estimated flux linkages are
determined, it is then a straight process to compute the rotor flux angle estimate rψθ
by;
⎟⎟⎠
⎞⎜⎜⎝
⎛= −
sdr
sqr
r ψψ
θψ ˆˆ
tanˆ 1 (3-9)
The block diagram of rotor flux angle estimation is illustrated in Figure 3-1.
51
)/(tan 1 sdr
sqr ψψ−
rψθ
sdsψsqsψ
rwsdsi
sqsi
sdsVs
qsV SpeedAdaptive
FluxObserver
Figure 3-1 The block diagram of rotor flux angle estimation
3.1.2. Adaptive Scheme for Speed Estimation
It can be seen that the state matrix of the observer ( A ) (given as (3.6) is a function
of the rotor speed. In a sensorless drive, the rotor speed must be estimated and fed
into A . The estimated rotor speed being denoted by rω , A becomes a function of
rω . It is important to note that the estimated rotor speed is considered as a parameter
in A ; however, in some other types of observers (e.g. extended Kalman filter), the
estimated speed is not considered as a parameter, but it is a state variable.
A speed adaptive flux observer algorithm is implemented by Kubota [5] on the
basis of an adaptive control scheme as shown in Figure 3-2. In this way, it is possible
to implement a speed estimator which estimates the electrical rotor speed of an
induction machine.
52
Tsqs
Vsds
Vss
V ⎥⎦⎤
⎢⎣⎡=
Tsqs
isds
iss
i ⎥⎦⎤
⎢⎣⎡=
srψ
A
rω
xAˆˆ
ssBV
pKK i
p +
x
0x
e
e
e
Ge
Figure 3-2 Adaptive state observer
To obtain error dynamics, (3-5) is subtracted from (3-2), yielding the following
observer-error equation;
[ ] [ ] [ ] ( )[ ]ss
ss
ss
ss iiGBVxABVAxx
dtdx
dtd ˆˆˆˆ −++−+=−
[ ] ( )[ ] [ ]xAeGCAedtd ˆΔ−+= (3-10)
where
( )xxe ˆ−=
53
[ ] ⎥⎦
⎤⎢⎣
⎡ΔΔ−
=ΔJ
cJA
r
r
ωω
0/0
( )m
rs
LLLc σ
=
rrr ωωω −=Δ ˆ
It can be understood that error dynamics are described by the eigenvalues of
GCA+ and these could also be used to design a stable observer (selecting an
appropriate gain matrix, G for stability). However, in order to determine the stability
of the error dynamics of the observer, it is possible to use Lyapunov stability
theorem, which gives a sufficient condition for the uniform asymptotic stability of a
non-linear system by using a Lyapunov function V which has to satisfy various
conditions. For instance, it must be continuous, differentiable, positive semi-definite,
etc. Such a function exists and the following Lyapunov function is introduced.
λωω /)ˆ( 2rr
T eeV −+= (3-11)
where λ is a positive constant. The time derivative of V becomes,
( ) ( ) λωωψψω /ˆ2/)ˆˆ(2 rrsdr
siqs
sqr
sidsr
TT
dtdceeeGCAGCAeV
dtd
Δ+−Δ−+++=
(3-12)
where,
sqs
sqs
siqs
sds
sds
sids
iie
iieˆ
ˆ
−=
−=
54
Since a sufficiency condition for uniform asymptotic stability is that the derivative
of Lyapunov function, dV/dt, is to be negative semi-definite. If observer gain matrix
G is selected appropriately, the first term of (3-10) becomes always negative semi-
definite, which can be satisfied by ensuring that the sum of the last two terms in (3-
10) is zero, so the observer is stable. As a result, we can come up with an adaptive
scheme for the speed estimation by equating the second term to the third term in (3-
10).
( ) ceedtd s
drsiqs
sqr
sidsr /ˆˆˆ ψψλω −= (3-13)
Thus, from the (3-13), the speed is estimated as;
( )dteeK sdr
siqs
sqr
sidsir ∫ −= ψψω ˆˆˆ (3-14)
However, so as to improve the performance of the speed observer, it is modified to
( ) ( )dteeKeeK sdr
siqs
sqr
sidsi
sdr
siqs
sqr
sidspr ∫ −+−= ψψψψω ˆˆˆˆˆ (3-15)
where pK and iK are arbitrary positive gains.
If the observer gain matrix G is chosen so that the A-GC term is negative semi-
definite, then the speed observer is stable. To ensure the stability at all speeds, the
conventional technique is to select observer poles which are proportional to motor
poles [27]. Thus, by using the pole placement technique, the gain matrix is obtained
as
⎥⎦
⎤⎢⎣
⎡++
−=JgIgJgIg
G423
221 (3-16)
55
The four gains in G are obtained from the eigenvalues of the induction machine as
follows,
224
221121112
3
222
22111
)1())(1())(1(
))(1())(1(
i
rrrr
i
rr
amcgaakcacamg
amgaamg
−−=+−−+−=
−=+−=
(3-17)
For DSP implementation the discretized form of the observer in (3-2) and the
adaptation mechanism (3-15) are used. Thus the discretized observer is described by
( ))(ˆ)()(ˆ)1(ˆ mimiGVBmxAmx ss
ssd
ssdd −++=+ (3-18)
where dA and dB are discretized matrices.
2)()exp(
2ATATIATAd ++≈= (3-19)
[ ]2
)exp(2
0
ABTBTBdtATBt
d +≈= ∫ (3-20)
The observer poles are chosen to be proportional to the poles of induction machine.
To make the scheme insensitive to the measurement noise, the constant m in (3-17) is
selected to be low. However, this pole-placement technique may have some
disadvantages and may not ensure to have a good observer dynamics. It requires
extensive computation time due to updating of matrix G and discretization
procedure, and this is a disadvantage. The observer dynamics can be adversely
affected by the fact that for small sampling time and low rotor speed, the discrete-
locus being very close to the stability limit and in case there are computational errors,
then an instability may arise. It is possible to overcome some of these difficulties.
56
For example, two different constant gain matrices G and G’ are predetermined and
used according to the rotor speed. (one for speed values less than a specified value
and the other for speed values higher than this specific value.)
At the experimental work, for simplifying the DSP implementation, the
proportionality constant k is chosen as 1.0 so G = 0 and initial value of the estimated
speed is zero.
The performance of the speed adaptive flux observer is improved in [28] in terms
of stability and accuracy at low speed region. In addition, in [5], adaptive stator
resistance estimation is achieved together with adaptive speed estimation.
3.2. Kalman Filter for Speed Estimation
R. E. Kalman has brought a recursive solution to the problem of discrete-time
filtering with his paper published in 1960. Rapid expansion and low cost of computer
systems has resulted in the use of Kalman filters in numerous research and projects
particularly on navigation and the modulation of servo systems.
Kalman filter consists of a number of mathematical equations which bring a
recursive solution to the well-known least-square method. The filter is a strong one
in the sense that it can estimate the past, present and also future states even in the
situations where the mathematical model used in the design does not totally reflect
the physical system.
3.2.1. Discrete Kalman Filter
Today, there are many derivatives of Kalman filter and many applications based on
these derivations. Among these derivatives, adaptive Kalman filters, non-linear
Kalman filters and discrete time Kalman filters are most widely-used ones. The
general operating principle lying behind all these derivations can be roughly
explained as the minimization of the total squares of error data. Kalman filter can
57
predict any data by the use of another data in a way to cause minimum error.Assume
that the mathematical method below is a state-space expression.
kkk
1k1k1kk
vxHzwuBxAx
+=++= −−− (3-21)
Here, x matrix stands for the state parameters of the system, A for system matrix, B
for input matrix, u for system input, z for system output, H for output matrix, w for
noise in the model (the noise affecting state parameters) and v for noise while
reading signals. A subscript under any parameter shows to which instant this
parameter belongs (for instance, kx shows the x value at k instant, 1−kx shows the x
value at k-1 instant which is just one step before k instant).
While forming discrete Kalman filter, it is assumed that w matrix, which is called
as process noise and includes the reaction differences between system model and real
system and the operational errors of the hardware on which the filter operates, and v
matrix, which is called as measurement noise and includes the errors in signal
measurement, have a mean value of zero and a normal distribution. The probability
expressions of these noise matrices are given in (3-22).
( ) ( )( ) ( )RNvp
QNwp
,0
,0
≈
≈ (3-22)
The matricesQ , called as process noise covariance, and R , called as measurement
noise covariance are generally considered as constant although they are unsteady. By
the use of the mathematical model (3-21) Kalman filter, a priori state estimate of the
system output and state parameters is made (3-24) and by the use of previous
estimations, P matrix, priori error covariance, is calculated (3-25). This matrix
includes the data concerning how valid the filter’s previous estimations are. The
estimation error can be found by comparing system output taken from the system and
the priori state estimate (3-23).
58
−−= kkk xHze ˆ (3-23)
The main goal of the filter is to correct the priori state estimate ( −kx ) by the use of
these error data and approximate to the real system output as much as possible. K
matrix (3-26), known as Kalman Gain, is used for this purpose. The found estimation
error (3-23) is multiplied by this gain and added to the priori state estimate. In this
way, state estimations are acquired (3-27). Later, posteriori estimate error covariance
is calculated so as to use at the next time step (3-28).
The operating principle of Kalman filter mentioned above can be expressed by the
mathematical equations given below. Priori state estimates are made.
11ˆˆ −−− += kkk uBxAx (3-24)
Priori error covariance is calculated.
QAPAP Tkk += −
−1 (3-25)
Kalman Gain is calculated.
( ) 1−−− += RHPHHPK Tk
Tkk (3-26)
State estimations are calculated.
( )−−+= kkkkk xHzKxx ˆˆˆ (3-27)
Posteriori estimate error covariance is calculated.
59
( ) −−= kkk PHKIP (3-28)
In order for the filter to operate, the first values of x and P matrices are required.
If the first values of these matrices can be measured, real values should be used. If
these matrices cannot be measured, the use of approximate values won’t affect the
filter performance, but at the beginning, it will lead to some wrong estimations for a
short time. P matrix should not be zero matrix because zero matrix means that
estimations are the same with the measurements, and in this case, estimations
become the same with the measurements at each time step. Generally, as the first
value of P matrix, identity matrix is used.
(3-24) and (3-25) are named as Time Update equations whereas the other three are
called as Measurement Update equations. By the use of these equations, the solution
can be summarized as in the Figure 3-3:
Figure 3-3 Discrete Kalman filter algorithm
60
The design of Kalman filter can be summarized as finding the mathematical
equations defining the system, that are Q and R matrices. Although the effect of the
mathematical model on filter performance is acute, very simple models may give
rather satisfactory results. The important thing is to express the dominant
characteristics of the system by means of the mathematical model.
R matrix can be found by measuring the noise in the measured signals. This
matrix can be obtained by putting the squares of the standard deviation of the noise
in the measured signals successively to the diagonal elements of R diagonal matrix
(3-29).
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
=
2
22
21
00
0000
m
R
σ
σσ
(3-29)
Q matrix includes the ambiguity assigned to the system model and it is expressed
through Equation 10 for a system model with three state parameters. In this equation,
iσ is the standard deviation value of the ith state parameter, and this value includes
the errors appearing upon the comparison of ith state parameter and real system and
the operating errors of the hardware in which the filter is operating.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=233231
322221
312121
σσσσσσσσσσσσσσσ
Q (3-30)
To get Q matrix is not as easy as R matrix. Most often, state parameters cannot be
measured and the operating errors to be faced in hardware cannot be predicted. Even
though the general way of obtaining this matrix is to make some assumptions and
61
come up with mathematical conclusions, it can also be found by means of trial and
error.
The smaller the values are for the values of standard deviation, the more the filter
relies on the system model. Even in the situations where the system model expresses
the real system quite well, giving very small values for the values of standard
deviation makes the improvement of filter parameters difficult.
On the other hand, the small values in R matrix show that the measurement is
reliable. Therefore, for the digital signals (in the case there is no sensor noise), this
value should be chosen very small. Very small values of this parameter affect the
filter performance badly by making the Kalman Gain almost constant. If it is chosen
as zero, then Kalman Gain turns out to be a constant matrix. In this case, constant
Kalman Gain leads P matrix, which includes estimation performance data, to be
zero and give the priori state estimate of the filter as output without correction.
3.2.2. “Obtaining “Synchronous Speed, ws” Speed Data through the use of
“Rotor Flux Angle”
It is evaluated that “Electrical synchronous speed” speed data can be attained by the
use of Rotor Flux Angle. For this purpose, constant acceleration mathematical model
has been used in the designed Kalman filter. Constant acceleration mathematical
model can be expressed in the state-space as follows:
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
−
k
k
k
k
1k
1k
1k2
k
k
k
001y ,100T10
2TT1
αωθ
αωθ
αωθ
(3-31)
Here, θ stands for “Rotor Flux Angle” angular position data, ω for “synchronous
speed, ws” angular speed data, α for angular acceleration data, T for sampling time
and y for system output. Since only θ angular position can be predicted through
62
“flux estimator”, only this state parameter has been used as system output. Constant
acceleration model does not include system dynamic. Assuming that system does not
reach to high speeds and it is linear, this model has been utilized. It is due to the
ignorance of the system dynamic that model shows weakness particularly at the high
frequencies (above 7 Hz).
When this mathematical model and the previous equations are applied,
“Synchronous speed, ws” angular speed data can be obtained by the use of “Rotor
Flux angle” θ angular position data. Due to the problems faced in the speed data
estimated via “Flux estimator”, the need for getting speed data from noisy position
data can be met with Discrete Kalman Filter Model.
3.2.3. Extended Kalman Filter (EKF)
The extended Kalman filter (EKF) could be used for the estimation of the rotor
speed of an induction machine. The EKF is suitable for use in high-performance
induction motor drives, and it can provide accurate speed estimates in a wide speed-
range, including very low speeds. It can also be used for joint state and parameter
estimation. However, it is computationally more intensive than both speed adaptive
flux observer and discrete kalman filter described in the previous sections.
The EKF is a recursive optimum stochastic state estimator which can be used for
the joint state and parameter estimation of a non-linear dynamic system in real-time
by using noisy monitored signals that are disturbed by random noise. This assumes
that the measurement noise and disturbance noise are uncorrelated. The noise sources
take account of measurement and modelling inaccuracies.
The EKF is a variant of the Kalman filter, but the extended version can deal with a
non-linear system. Once the differences between speed adaptive flux observer and
EKF are considered, it is noted that in the speed adaptive flux observer, the noise has
not been considered. So, it is a deterministic observer in contrast to the EKF which is
a stochastic observer. Furthermore, in the speed adaptive flux observer, the speed is
considered as a parameter, but in the EKF it is considered as a state. Similar to the
63
speed-adaptive flux observer, where the state variables are adapted by the gain
matrix (G), in the EKF the state variables are adapted by the Kalman gain matrix (K).
In a first stage of the calculations of the EKF, the states are predicted by using a
mathematical model of the induction machine (which contains previous estimates)
and in the second stage, the predicted states are continuously corrected by using a
feedback correction scheme. This scheme makes use of actual measured states by
adding a term to the predicted states (which are obtained in the first stage). The
additional term contains the weighted difference of the measured and estimated
output signals. Based on the deviation from the estimated value, the EKF provides an
optimum output value at the next input instant.
In an induction motor drive, the EKF can be used for the real-time estimation of the
rotor speed, but it can also be used for joint state and parameter estimation. For this
purpose, the stator voltages and currents are measured (or the stator voltages are
recontructed from the d.c. link voltage and the inverter switching signals) and the
speed of the machine can be obtained by the EKF quickly and precisely.
The main design steps for a speed-sensorless induction motor drive implementation
using the discretized EKF algorithm are as follows [26]:
1. Selection of the time-domain machine model;
2. Discretization of the induction machine model;
3. Determination of the noise and state covariance matrices Q, R, P;
4. Implementation of the discretized EKF algorithm; tuning.
These steps are now discussed.
3.2.3.1. Selection of the Time-Domain Machine Model
For the purpose of using EKF for the estimation of rotor speed of an induction
machine, it is possible to use various machine models. For example, it is possible to
use the equations expressed in the rotor-flux-oriented reference frame (ωg = ωmr), or
in the stationary reference frame. Due to convenience and computational reduction,
induction machine model at stationary reference frame is chosen. The main
advantages of using the model in the stationary reference frame are reduced
64
computation time (e.g. due to reduced non-linearities), smaller sampling times and
more stable behaviour. [26]
The two-axis state-space equations including rotor speed of the induction machine
in the stationary reference are as follows,
[ ] BuAxxdtd
+= (3-32)
where
[ ]Trsqr
sdr
sqs
sds iix ωψψ= , [ ]Ts
qss
ds VVu = ,
T
s
sL
L
B
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
=
000000
/100/1σ
σ
and
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−
−−
=
000000/1/00/10/0)()/(/100)/()(0/1
*
*
rrrm
rrrm
rrsmrsmrs
rsmrrrsms
TTLTTL
TLLLLLLTLLLTLLLT
A
ωω
σσωσωσ
)/())/((/1 2*srmrss LLLRRT σ+=
Then, the (3-32) takes the form below,
65
ss
sqs
sds
r
sqr
sdr
sqs
sds
r
sqr
sdr
sqs
sds
BVAx
V
VB
i
i
Ai
i
dtd
+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
ωψ
ψ
ωψ
ψ
(3-33)
and
xC ⋅=ssi (3-34)
xDsr ⋅=ψ (3-35)
where
[ ]Tsqs
sdss iii = , [ ]Ts
qrsdr
sr ψψψ = , [ ]Ts
qss
dss
s VVV =
⎥⎦
⎤⎢⎣
⎡=
00100001
C and ⎥⎦
⎤⎢⎣
⎡=
10000100
D
Note that, the rotor speed derivative has been assumed to be negligible, dωr/dt = 0.
Although this last equation implies that the machine has infinite inertia and therefore
unable to accelerate, actually this is not true. This is corrected by the operation of
Kalman filter (by the system noise, which also takes account of the computational
inaccuracies). Furthermore, it should be noted that the effects of saturation of the
magnetic paths of the machine have been neglected. This assumption is justified
since it can be shown that the EKF is not sensitive to changes in the inductances,
since changes in the stator parameters are compensated by the current loop inherent
in the EKF. The application of (3-32) in the EKF will give not only the rotor speed
66
but also the rotor flux-linkage components (and as a consequence the angle and
modulus of the rotor flux-linkage space vector will also be known). This is useful for
high-performance drive implementations. It is important to emphasize that the rotor-
speed has been considered as a state variable and the system matrix A is non-linear –
it contains the speed, A = A(x).
3.2.3.2. Discretization of the Induction Motor Model
For digital implementation of the EKF, the discretized machine equations are
obtained as follows:
)()()1( muBmxAmx dd +=+ (3-36)
where dA and dB are discretized system and input matrices respectively.
2)()exp(
2ATATIATAd ++≈= (3-37)
[ ]2
)exp(2
0
ABTBTBdtATBt
d +≈= ∫ (3-38)
By considering the system noise v(k) (v is the noise vector of the states), which is
assumed to be zero-mean, white Gaussian, which is independent of x(k), and which
has covariance matrix Q, the system model becomes:
)()()()1( kvmuBmxAmx dd ++=+ (3-39)
By considering a zero-mean, white Gaussian measurement noise, w(k) (noise in the
measured stator currents), which is independent of y(k) and v(k) and whose
covariance matrix is R, the output equation becomes,
67
)()()( kwmCxmy += (3-40)
3.2.3.3. Determination of the Noise and State Covariance Matrices Q, R, P
The goal of the Kalman filter is to obtain estimates about the unmeasurable states
(e.g. rotor speed) by using measured states, and also statistics of the noise and
measurements. In general, by means of the noise inputs, it is possible to take account
of computational inaccuracies, modelling errors, and errors in the measurements. The
filter estimation ( x ) is obtained from the predicted values of the states (x) and this is
corrected recursively by using a correction term, which is the product of the Kalman
gain (K) and the deviation of the estimated measurement output vector and the actual
output vector ( y- y ). The Kalman gain is chosen to result in the best possible
estimated states. Thus, the filter algorithm contains basically two main stages, a
prediction stage and a filtering stage as illustrated at Figure 3-3.
During the prediction stage, the next predicted values of the states x(k+1) are
obtained by using a mathematical model (state-variable equations) and also the
previous values of the estimated states. Furthermore, the predicted state covariance
matrix (P) is also obtained before the new measurements are made, and for this
purpose, the mathematical model and also the covariance matrix of the system (Q)
are used.
In the second stage, which is the filtering stage, the next estimated states, x ( k+1)
are obtained from the predicted estimates x(k+1) by adding a correction term K( y-
y ) to the predicted value. This correction term is a weighted difference between the
actual output vector (y) and the predicted output vector ( y ), where K is the Kalman
gain. Thus, the predicted state estimate (and also its covariance matrix) is corrected
through a feedback correction scheme that makes use of the actual measured
quantities.
The Kalman gain is chosen to minimize the estimation-error variances of the states
to be estimated. The computations are realized by using recursive relations. The
68
algorithm is computationally impressive, and the accuracy also depends on the model
parameters used. A critical part of the design is to use correct initial values for the
various covariance matrices. These can be obtained by considering the stochastic
properties of the corresponding noises. Since these are usually not known, in most
cases they are used as weight matrices, but it should be noted that sometimes simple
qualitative rules can be set up for obtaining the covariances in the noise vectors. With
the advances in DSP technology, it is possible to conveniently implement an EKF in
real time.
The system noise matrix Q is a five-by-five matrix, the measurement noise matrix
R is a two-by-two matrix, so in general this would require the knowledge of 29
elements. However, by assuming that the noise signals are not correlated, both Q and
R are diagonal, and only 5 elements must be known in Q and 2 elements in R.
However, the parameters in the direct and quadrature axes are the same, which
means that the first two elements in the diagonal of Q are equal (q11 = q22), the third
and fourth elements in the diagonal Q are equal (q33 = q44), so Q = diag(q11, q11, q33,
q33, q55) contains only 3 elements which have to be known. Similarly, the two
diagonal elements in R are equal (r11 = r22 = r), thus R = diag(r, r). It follows that in
total only 4 noise covariance elements must be known.
3.2.3.4. Implementation of the Discretized EKF Algorithm; Tuning
As discussed above, the EKF algorithm contains basically two main stages, a
prediction stage and a filtering stage. During the prediction stage, the next predicted
values of the states x(k+1) [which will be denoted by x* (k+1)] and the predicted
state covariance matrix (P) [which will be denoted by P*] are also obtained. For this
purpose, the state-variable equations of the machine and the system covariance
matrix (Q) are used. During the filtering stage, the filtered states ( x ) are obtained
from the predicted estimates by adding a correction term to the predicted value (x*);
this correction term is Ke = K(y- y ), where e = ( y- y ) is an error term, and it uses
69
measured stator currents, y = iss, y = îss. This error is minimized in the EKF. The
EKF equation is given as,
( )ss
ss
ss iiKBVxxAx
dtd ˆˆ)ˆ(ˆˆ −++= (3-41)
The structure of the EKF is shown in Figure 3-4. The state estimates are obtained
by the EKF algorithm in the following seven steps:
Step 1: Initialization of the state vector and covariance matrices
Starting values of the state vector )( 00 txx = and the starting values of the noise
covariance matrices Q0 (diagonal 5 x 5 matrix) and R0 (diagonal 2 x 2 matrix) are set,
together with the starting value of the state covariance matrix P0 (which is a 5 x 5
matrix), where P is the covariance matrix of the state vector.
70
B
B
∫
∫
C
C
+
+
+++
+
++
K
v w
e
+_
u = u x x
xx
A(x)
A(x)
is
is
x0
Induction motor
Figure 3-4 The structure of EKF algorithm
The initial-state covariance matrix can be considered as a diagonal matrix, where
all the elements are equal. The initial values of the covariance matrices reflect on the
degree of knowledge of the initial states. A suitable selection allows us to obtain
satisfactory speed convergence, and avoids divergence problems or unwanted large
oscillations.
Step 2: Prediction of the state vector
Prediction of the state vector at sampling time (k+1) from the input u(k), state
vector at previous sampling time x (k), by using Ad and Bd is obtained by performing
71
)()(ˆ)1(* kuBkxAkx dd +=+ (3-42)
The notation x*(k+1) P*(k+1) and etc. mean that it is a predicted value at the (k+1)-
th instant, and it is based on measurements up to the kth instant.
Step 3: Covariance estimation of prediction
The covariance matrix of prediction is estimated as,
QkfkPkfkP T +++=+ )1()(ˆ)1()1(* (3-43)
where P (k) denotes prediction at time k based on data up to time k and f is the
following gradient matrix:
)1(ˆ)()1( +=+∂∂
=+ kxxdd uBxAx
kf (3-44)
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−
−−−−
=+
10000/1/0
/10/)()()/(/10
)()/()(0/1
)1(
*
*
sdrrrrm
sqrrrrm
sdrrsmrrsmrsmrs
sqrrsmrsmrrrsms
TTTTTTLTTTTTTL
LLTLTLLTLLLTLTTLLTLLLTLTLLTLTT
kfψωψω
ψσσσωψσσωσ
(3-45)
where ωr = ω r(k+1), ψsdr = ψ s
dr(k+1), ψsqr = ψ s
qr(k+1).
Step 4: Kalman filter gain computation
The Kalman filter gain (correction matrix) is computed as,
[ ] 1** )1()1()1()1()1()1( −++++++=+ RkhkPkhkhkPkK TT (3-46)
72
where h(k+1) is a gradient matrix and defined as,
⎥⎦
⎤⎢⎣
⎡=+
∂∂
=++=
0001000001
)1(
)()1( )1(*
kh
Cxx
kh kxx
(3-47)
For the induction machine application, the Kalman gain matrix (K) contains two
columns and five rows.
Step 5: State-vector estimation
The state-vector estimation (corrected state-vector estimation, filtering) at time
(k+1) is performed as:
[ ])1(ˆ)1()1()1()1(ˆ * +−++++=+ kykykKkxkx (3-48)
Where
)1()1(ˆ * +=+ kCxky (3-49) Step 6: Covariance matrix of estimation error
The error covariance matrix can be obtained from
)1()1()1()1()1(ˆ ** +++−+=+ kPkhkKkPkP (3-50)
Step 7: Put k = k+1, x(k) = x(k-1), P(k) = P(k-1) and go to Step 1.
The EKF described above can be used under both steady-state and transient
conditions of the induction machine for the estimation of the rotor speed. By using
73
the EKF in the drive system, it is possible to implement a PWM inverter-fed
induction motor drive without the need of an extra speed sensor. It should be noted
that accurate speed sensing is obtained in a very wide speed-range, down to very low
values of speed (but not zero speed). However, care must be taken in the selection of
the machine parameters and covariance values used. The speed estimation scheme
requires the monitored stator voltages and stator currents. Instead of using the
monitored stator line voltages, the stator voltages can also be reconstructed by using
the d.c. link voltage and inverter switching states, but especially at low speeds it is
necessary to have an appropriate dead-time compensation, and also the voltage drops
across the inverter switches (e.g. IGBTs) must be considered.
The tuning of the EKF involves an iterative modification of the machine
parameters and covariances in order to yield the best estimates of the states.
Changing the covariance matrices Q and R affects both the transient duration and
steady-state operation of the filter. Increasing Q corresponds to stronger system
noises, or larger uncertainity in the machine model used. The filter gain matrix
elements will also increase and thus the measurements will be more heavily weighted
and the filter transient performance will be faster. If the covariance R is increased,
this corresponds to the fact that the measurements of the currents are subjected to a
stronger noise, and should be weighted less by the filter. Thus the filter gain matrix
elements will decrease and this results in slower transient performance. Finally, it
should be noted that in general, the following qualitative tuning rules can be
obtained:
Rule 1: If R is large then K is small (and the transient performance is faster).
Rule 2: If Q is large then K is large (and the transient performance is slower).
However, if Q is too large or if R is too small, instability can arise.
It is possible to derive similar rules to these rules, and to implement a fuzzy-logic-
assisted system for the selection of the appropriate covariance elements.
In summary it can be stated that the EKF algorithm is computationally more
intensive than the algorithm for the full-order state observer described in the previous
section. The EKF can also be used for joint state and parameter estimation. It should
74
be noted that in order to reduce the computational effort and any steady state error, it
is possible to use various EKFs, which utilize reduced-order machine models and
different reference frames.
75
CHAPTER 4
4.1 Experimental Work
4.1.1 Induction Motor Data
A squirrel cage induction motor with electrical name plate data shown in Table 4-1
has been used in experiments.
Table 4-1 Induction motor electrical data
SIEMENS 1LA7107-4AA1
Manufacturer
Specification
Frequency 50 Hz
Nominal Voltage 400 Vrms
Nominal Current 6.4 Arms
Nominal Power 3 kW
Power Factor 0.82
Nominal Speed 1420 rpm
Nominal Torque 20 Nm
Motor Inertia 0.05 kgm2
Number of Poles 4
4 SIMULATIONS AND EXPERIMENTAL WORK
76
Parameters of the induction motor are needed to be known in implementing the
speed adaptive flux observer. These are shown in Table 4-2.
Table 4-2 Induction motor parameters
SIEMENS 1LA7107-4AA1
Test Result Manufacturer
Specification
Rotor resistance per phase
(referred)
2.19 Ω 1.658 Ω
Stator resistance per phase 1.80 Ω 2.037 Ω
Stator self inductance per phase 0.192 H 0.24268 H
Rotor self inductance per phase 0.192 H 0.23379 H
Magnetizing inductance 0.184 H 0.22885 H
The per unit values related to direct on-line starting are given at Table 4-3.
Manufacturer provided direct on-line starting curves for the induction motor are
shown in Figure 4-1 - Figure 4-3.
Table 4-3 Direct on-line starting per unit definitions
SIEMENS 1LA7107-4AA1
Manufacturer
Specification
Per Unit Speed 1420 rpm
Per Unit Torque 14.8 Nm
Per Unit Power 2.2 kW
77
Table 4-3 (Cont’d)
Moment of Inertia at Start-up 0.05 kgm2
Input Voltage 400 V
Frequency 50 Hz
The direct on-line starting moment versus motor speed graph is illustrated at Figure
4-1.
Figure 4-1 The direct on-line starting moment vs motor speed graph
The direct on-line starting motor current versus motor speed at given load
condition is given at Figure 4-2.
78
Figure 4-2 The direct on-line starting motor current vs motor speed graph
The direct on-line starting time versus motor speed at given load condition is given
at Figure 4-3. The per unit valuıes related to direct on-line starting are given at Table
4-3. Manufacturer provided direct on-line starting curves for the induction motor
which are shown in Figure 4-1 - Figure 4-3
Figure 4-3 The direct on-line starting time vs motor speed graph
79
4.1.2. Experimental Set-up
The schematic block diagram of the experimental set-up is illustrated at Figure 4-4.
Figure 4-4 The schematic block diagrams of experimental set-up
The rectifier used in this drive is bridge rectifier which is 450V, 28A that consists
of six uncontrolled diodes and produced by IXYS. The three-phase voltage is
supplied over a digitally controlled three-phase supply.
The rectified output voltage is filtered by two dc-link capacitors each being 4700
μF (400V) and connected in series. 600 KΩ, 0.6 W resistors are connected across
each capacitor for proper voltage sharing.
The voltage across the capacitors is raised by charging them over a 100 Ω soft start
resistor to limit the in-rush current at starting. Upon the capacitors are charged to a
predefined level, a relay shorts the two ends of the resistor so that the rectifier output
voltage is applied directly onto the capacitors.
80
A motor drive electronics card is used to drive inverter circuit IGBTs. This drive
circuit incorporates an IGBT voltage source inverter, an IGBT gate driver, phase
current sensors, a dc-link voltage sensor, DC/DC converters which are needed by
control electronics and signal interface adaptation circuits.
The inverter used in the drive system is Semikron IGBT module (SKM 40 GDL
123 D) with rated values of 1200V and 40 A. IGBTs in this module are driven by a
gate drive electronic module, Concept Scale Driver (Scale Driver 6SD106E). The
module provides over-current and short-circuit protection for all six IGBTs in the full
bridge by real-time tracking of the collector-emitter voltage of the switches.
The dc-link voltage is measured with a voltage sensor (LEM LV25-P) at the motor
drive electronics card. The magnitude of the dc-link voltage is measured to
reconstruct the phase voltages in the control software with the information of PWM
cycles.
The other measured variables are stator phase currents. For this purpose (LEM LA
25-NP), current transducers are used. These sensors are capable of sensing AC, DC
and mixed current waveforms. The sensor has multi-range current sensing options
depending on the pin connections. The sensors use hall-effect phenomena to sense
the current. The output of these sensors is between ±15V and unipolar.
The PWM signals generated by DSP are amplified at the motor drive electronics
card to make them compatible with the gate drive module inputs. Moreover, the
errors (gate drive card errors such as short-circuit error, over-current error, and an
external interrupt) are monitored in order to stop IGBT operation.
In order to verify the estimator performance of the induction motor, an incremental
encoder is mounted on the rear side of the shaft to determine real rotor speed. The
encoder is 1440 pulses/rev (Us Digital Corp. E2 – Optical Kit Encoder).
The real-time experiments have been carried out by the use of electronic control
card including mainly TI TMS320F2812 digital signal processor, XILINX
XCS2S150E Field Programmable Gate Array (FPGA), various chips which enable
transformations from analog to digital or vice versa and supplementary circuits. In
order to run the real-time control algorithm and create PWM signals, Texas
81
Instruments’ TMS320 processor is used in this work. The F2812 is a member of the
“C2000 DSP” platform, and is optimized specifically for motor control applications.
It uses a 16-bit word length along with 32-bit registers. The F2812 has application-
optimized peripheral units, coupled with the high-performance DSP core, enables the
use of advanced control techniques for high-precision and high-efficiency full
variable-speed control of motors. The event managers of F2812 include special
pulse-width modulation (PWM) generation functions such as a programmable dead-
band function and a space-vector PWM state machine for 3-phase motors that also
provides a quite high efficiency in the switching of power transistors, quadrature
encoder pulse circuit module to read encoder signals. F2812 also contains 16
channels, 12-bit A/Ds, enhanced controller area network (eCAN), serial
communication interface (SCI) and general purpose digital I/Os (GPIO) as
peripherals. XILINX XCS2S150E Field Programmable Gate Array (FPGA) is used
for logical operations and external memory access.
Magtrol test bench is used for applying load torque. Magtrol test bench includes
hysteresis load, torque analyser, power analyzer, signal amplifier and graphical user
interfece. The load characteristics of magtrol test bench are tuned in order to ensure
proper load characteristics. The tuned load response is given at Figure 4-5.
82
Figure 4-5 The Magtrol test bench load response
The pictures of experimental set-up are given at Figure 4-6.
83
Figure 4-6 The experimental set-up
4.1.3 Experimental Results of Speed Adaptive Flux Observer
The induction motor parameters at Table 4-2 are used at the experimental stage of
the speed adaptive flux observer.
The speed estimator designed with speed adaptive flux observer has been tested
experimentally for satisfactory operation at different speeds of motor.
The control parameters used at the speed adaptive flux observer experiments are
listed at Table 4-4.
84
Table 4-4 Control parameters used at the speed adaptive flux observer experiments
KI 194 *e
dsI Current Regulator KP 1
KI 1 *e
qsI Current Regulator KP 194
KI 0.08
rω Speed Requlator KP 0.005
KI 800 Adaptive Scheme Gain
KP 4
4.1.3.1 No-Load Experiments of Speed Adaptive Flux Observer
In these no-load experiments, motor is run in the closed- loop speed mode and the
quadrature encoder coupled to the shaft of the motor is utilized in order to verify the
estimated speed. Log of the mechanical rotor angle, the estimated speed, and the
actual speed are taken for 50rpm, 100rpm, 500rpm, 1000rpm and 1500rpm constant
speed request. The speed estimate derived from actual rotor (quadrature encoder)
position is given as dotted line.
Speed command, real rotor speed and estimated rotor speed for 50rpm, 100 rpm,
500 rpm 1000 rpm and 1500 rpm cases are respectively represented at Figure 4-7 to
Figure 4-11.
85
Figure 4-7 50 rpm speed reference, motor speed estimate
Figure 4-8 100 rpm speed reference, motor speed estimate
86
It can be seen from Figure 4-7 and Figure 4-8, no-load speed estimation
performance of speed adaptive flux observer is not satisfactory at low speeds such as
50 rpm and 100 rpm. Also, it is observed that once the speed is increased to 100 rpm,
speed estimation error percentage decreases from 40 % to 10%. By considering
equations (2-37) and (3-15), the speed estimate of speed adaptive flux observer is
derived from torque error representation. Since the torque error is very small at no-
load case, the speed estimates at no-load has lower performance. This results in
performance decrease due to the poor flux estimator characteristic at low speeds. In
addition to that, it is particularly at lower speeds that motor parameter variations
have significant influence on steady state and dynamic performance of the drive
system.
Figure 4-9 500 rpm speed reference, motor speed estimate
87
Figure 4-10 1000 rpm speed reference, motor speed estimate
Figure 4-11 1500 rpm speed reference, motor speed estimate
88
It can be seen from Figure 4-9 to Figure 4-11 that no-load speed estimation
performance of speed adaptive flux observer is satisfactory at speeds higher than 100
rpm. It is observed from Figure 4-9 to Figure 4-11. The observed speed estimator
error is less than 1 % at at speeds higher than 100 rpm.
In Figure 4-13 to Figure 4-12, quadrature encoder position, phase currents, and
voltages are given for 50rpm case.
Figure 4-12 50 rpm speed reference, motor quadrature encoder position
Figure 4-13 50 rpm speed reference, motor phase currents
89
Figure 4-14 50 rpm speed reference, motor phase voltages
The figures from Figure 4-9 to Figure 4-11 demonstrate that stator phase currents
and phase voltages are sinusoidal and the quadrature encoder position data are
consistent with the estimated speed.
4.1.3.2. The Speed Estimator Performance under Switched Loading
In this section, the performance of the drive system is investigated under switched
loading. The loading is obtained by using the Magtrol dynamometer coupled to the
shaft of the induction motor.
The Magtrol load dynamometer has following properties:
Tmax = 56 Nm,
P = 8 kW
Load experiments are at 50rpm, 100 rpm, 500rpm, 1000rpm and 1500rpm constant
speed references. Load experiments are carried out with the same conditions;
90
however, the load is switched on and off. The Table 4-5 shows the references and
corresponding measurements under loading.
Table 4-5 Loading measurements
Speed Reference
(rpm)
Real Rotor Speed
(rpm)
Estimated Speed
(rpm)
Load
Tload(Nm)
50 46.19 50.52 10
100 102.7 100.4 10
500 503.4 499.9 10
1000 999.8 1000 10
1500 1499 1500 8
Speed command, real rotor speed and estimated rotor speed for 50rpm, 100 rpm,
500 rpm 1000 rpm and 1500 rpm cases at Figure 4-15, Figure 4-17, Figure 4-19,
Figure 4-21 and Figure 4-23 respectively. The load profiles are added to speed
response at Figure 4-16, Figure 4-18, Figure 4-20, Figure 4-22 and Figure 4-24.
91
Figure 4-15 50 rpm speed reference, motor speed estimate under switched loading-1
Figure 4-16 50 rpm speed reference, motor speed estimate under switched loading-2
92
Figure 4-17 100 rpm speed reference, motor speed estimate under switched loading-1
It can be observed in Figure 4-15 and Figure 4-17 that the speed estimator
performance under loading is increased when they are compared with no–load test
results in Figure 4-7 and Figure 4-8. The performance increase is enabled due to
significant torque error component. So, more proper speed estimation is achieved
under loading at low speeds, but dynamic performance is still unsatisfactory.
93
Figure 4-18 100 rpm speed reference, motor speed estimate under switched loading-2
Figure 4-19 500 rpm speed reference, motor speed estimate under switched loading-1
94
Figure 4-20 500 rpm speed reference, motor speed estimate under switched loading-2
Figure 4-21 1000 rpm speed reference, motor speed estimate under switched loading-
1
95
Figure 4-22 1000 rpm speed reference, motor speed estimate under switched loading-
2
Figure 4-23 1500 rpm speed reference, motor speed estimate under switched loading-
1
96
Figure 4-24 1500 rpm speed reference, motor speed estimate under switched loading-
2
At the experiments of speed adaptive flux observer, the speed variation due to
switched loading is quite small and the drive system quickly reaches to the steady
state. The load switching times can be seen from speed graphs for time as speed
decreases.
It can be deduced from the experiments that the speed estimator performance under
loading are better than no–load test results.The performance increase is enabled due
to significant torque error component. Hence, more proper speed estimation is
achieved under loading at low speeds.
97
4.1.3.3. The Speed Estimator Performance under Accelerating Load
The speed estimator performance of the speed adaptive flux observer is
investigated under accelerating torque. The aim of this section is to ensure sensorless
vector drive performance while accelerating load.
Speed command, real rotor speed and estimated rotor speed for 50rpm to 500rpm,
500 rpm to 750rpm and 1000 to 1250 rpm cases are represented at Figure 4-25,
Figure 4-27 and Figure 4-29 respectively. The load profiles are added to speed
response at Figure 4-26, Figure 4-28 and Figure 4-30.
It can be deduced from Figure 4-25 to Figure 4-30 that the acceleration under
loading could be achieved by using speed adaptive flux observer algorithm.
Figure 4-25 50 rpm to 500 rpm speed reference, motor speed estimate under
accelerating load-1
98
Figure 4-26 50 rpm to 500 rpm speed reference, motor speed estimate under
accelerating load-2
Figure 4-27 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-1
99
Figure 4-28 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-2
Figure 4-29 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-1
100
Figure 4-30 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-2
4.1.3.4. The Speed Estimator Performance under No-Load Speed Reversal
The speed estimator performance of the speed adaptive flux observer is observed
under no-load speed reversal. The aim of this section is to determine the performance
of the speed estimator at low speeds and at zero speed while the machine is changing
the direction of rotation.
Figures 4-31 – 4-33 show the speed command, real rotor speed and estimated rotor
speed for different speeds. Fig.4-31 is for the operation of the machine at 50 rpm, in
transition from 50 rpm to -50 rpm, and at -50 rpm.. Fig.4-32 is for the operation of
the machine at 500 rpm, in transition from 500 rpm to -500 rpm, and at -500 rpm.
Fig.4-33 is for the operation of the machine at 1000 rpm, in transition from 1000 rpm
to -1000 rpm, and at -1000 rpm.
101
Figure 4-31 The speed reference, motor speed estimate under no-load speed reversal
for the speed range 50 rpm to -50 rpm
Figure 4-32 The speed reference, motor speed estimate under no-load speed reversal
for the speed range 500 rpm to -500 rpm
102
Figure 4-33 The speed reference, motor speed estimate under no-load speed reversal
for the speed range 1000 rpm to -1000 rpm
As it can be seen from Figure 4-31, Figure 4-32 and Figure 4-33, both at low
speeds and the zero speed crossing, speed estimator performance decreases due to the
flux estimator characteristic. It should be kept in mind that phase voltages are input
for speed adaptive flux estimator method and the phase voltages are almost zero at
the zero speed crossing which leads the mathematical model of induction motor to be
unobservable. So the speed estimation at zero speed is not possible by enabling speed
adaptive flux observer.
The experiments of speed adaptive flux observer show that speed estimator based
on speed adaptive flux observer has very high tracking capability for whole speed
range and for no-load and with load cases. However, the speed-loop performance, the
flux and the speed estimation accuracies should be improved for whole loading range
and for low speeds. Improvement could be achieved by increasing the computational
load of the processor. Some improvements are suggested at [28]
103
4.1.4 Experimental Results of Kalman Filter for Speed Estimation
The induction motor parameters at Table 4-2 are used at the experimental stage of
Kalman filter for speed estimation.
The speed estimator designed with Kalman filter for speed estimation algorithm
has been tested experimentally for satisfactory operation at different speeds of motor.
The control parameters used at the speed adaptive flux observer experiments are
listed at Table 4-6.
Table 4-6 Control parameters used at Kalman filter for speed estimation experiments
KI 49 *e
dsI Current Regulator KP 0.5
*eqsI Current Regulator KI 49
KP 0.5
KI 0.08 rω Speed Requlator
KP 0.005
KI 800 Adaptive Scheme Gain
KP 4
4.1.4.1. No-Load Experiments of Kalman Filter for Speed Estimation
In the no-load experiment, motor runs in the closed-loop speed control mode and
the quadrature encoder coupled to the shaft of the motor is utilized in order to verify
the estimated speed. The output of Kalman filter for speed estimation is utilized as
104
speed feedback. Logs of the mechanical rotor angle, the estimated and the actual
speeds are kept for constant speed requests of 50rpm, 100rpm, 250 rpm, 500rpm,
1000rpm and 1500rpm. The speed estimate derived from actual rotor (quadrature
encoder) position is given as dotted line.
Figure 4-34 to Figure 4-39 show the time variations of the set speed, the real and
the estimated rotor speeds for 50rpm 100 rpm, 250rpm, 500 rpm 1000 rpm and 1500
rpm. .
Since Kalman filter for the speed observer requires the use of the estimated flux
data derived from the speed adaptive flux estimator, the output performance of the
Kalman estimator is bounded by the accuracy in the flux estimation and the accuracy
of the speed observer. Since the adaptive scheme is internally corrected by speed
estimation of the flux observer, there is no direct correlation between speed adaptive
flux estimation scheme and Kalman filter for speed estimation. So, closed-loop
adaptive scheme is not statisfied. In other words, the flux observer is independent of
the closed-loop speed feedback which may lead to poorer performance of Kalman
filter for speed estimation compared to the estimated speed output of the speed
adaptive flux observer.
105
Figure 4-34 50 rpm speed reference, motor speed estimate
Figure 4-35 100 rpm speed reference, motor speed estimate
106
Figure 4-36 250 rpm speed reference, motor speed estimate
As it can be seen in Figure 4-34 - Figure 4-36, the no-load closed-loop speed
estimation performance of the Kalman filter state observer is poorer than the
performance of the speed adaptive flux observer. The speed curves at start up are
more oscillatory, and the steady state offset is higher than that in the speed adaptive
flux observer.
It is observed that once the speed increases the percentage error state in the speed
estimation decreases at steady-state. Note that the percentage speed estimation errors
at steady-state for 50 rpm and 100 rpm speed commands in Figs.4-34 and 4-35 are
60% and 31%, respectively.
107
Figure 4-37 500 rpm speed reference, motor speed estimate
Figure 4-38 1000 rpm speed reference, motor speed estimate
108
Figure 4-39 1500 rpm speed reference, motor speed estimate
It can be seen from Figure 4-37 to Figure 4-39 that no-load speed estimation
performance of Kalman state observer is increased at speeds higher than 250 rpm. It
is observed from Figure 4-37 to Figure 4-39 that speed estimator error percentage is
less than 3 % at at speeds higher than 250 rpm.
From Figure 4-40 to Figure 4-42, quadrature encoder position, phase currents and
voltages are given for 50rpm case.
109
Figure 4-40 50rpm speed reference, motor quadrature encoder position
Figure 4-41 50 rpm speed reference, motor phase currents
110
Figure 4-42 50 rpm speed reference, motor phase voltages
Figure 4-40 to Figure 4-42 demonstrate that the stator phase currents and phase
voltages are sinusoidal and the quadrature encoder position data are consistent with
the estimated speed.
4.1.4.2. The Kalman Filter for Speed Estimation Performance under
Switched Loading
This section investigates the performance of the drive system under switched
loading conditions. The loading pattern is obtained by using the Magtrol
dynamometer coupled to the shaft of the induction motor.
Figures 4-43 – 4-51 show the speed command, real rotor speed and estimated rotor
speed for different speeds. Figures are for the operation of the machine at 50, 100,
500, 1000, and 1500 rpm, respectively. The load profiles are added to the speed
responses in Figure 4-44, Figure 4-46, Figure 4-48, Figure 4-50 and Figure 4-52.
111
Figure 4-43 50 rpm speed reference, motor speed estimate under switched loading-1
Figure 4-44 50 rpm speed reference, motor speed estimate under switched loading-2
112
As it is observed from Figure 4-43 and Figure 4-44, there isn’t proper speed control
at 50 rpm. So, closed-loop speed control could not be achieved at 50rpm by enabling
Kalman filter state observer.
Figure 4-45 100 rpm speed reference, motor speed estimate under switched loading-1
113
Figure 4-46 100 rpm speed reference, motor speed estimate under switched loading-2
Figure 4-47 500 rpm speed reference, motor speed estimate under switched loading-1
114
Figure 4-48 500 rpm speed reference, motor speed estimate under switched loading-2
Figure 4-49 1000 rpm speed reference, motor speed estimate under switched loading-
1
115
Figure 4-50 1000 rpm speed reference, motor speed estimate under switched loading-
2
Figure 4-51 1500 rpm speed reference, motor speed estimate under switched loading-
1
116
Figure 4-52 1500 rpm speed reference, motor speed estimate under switched loading-
2
No connection could be established between the speed estimator performance and
loading. It seems that loading do not have any dominant effect on the speed
estimation characteristics of Kalman filter state observer. It is hard to conclude from
the experimental result that loading and speed estimator performances have a
correlation. Closed-loop operation could be achieved at speeds higher than 50 rpm.
4.1.4.3. The Kalman Filter for Speed Estimation Performance under
Accelerating Load
The section investigates the performance of the Kalman filter state observer for
speed estimation under accelerating torque. The aim in this investigation is to ensure
sensorless vector drive performance while accelerating under loading. Note that
Kalman filter used for the speed estimation does not include the dynamics of the
system. Although this corresponds to infinite inertia, actually this is not true, but the
117
required correction is realized by the Kalman filter as the system noise, which also
takes account of the computational inaccuracies.
Speed command, real rotor speed and estimated rotor speed for 100rpm to 250rpm,
500 rpm to 750rpm and 1000 to 1250 rpm cases are represented at Figure 4-53,
Figure 4-55 and Figure 4-57.respectively. The load profiles are added to speed
response at Figure 4-54, Figure 4-56 and Figure 4-58.
Figure 4-53 100 rpm to 250 rpm speed reference, motor speed estimate under
accelerating load-1
118
Figure 4-54 100 rpm to 250 rpm speed reference, motor speed estimate under
accelerating load-2
Figure 4-55 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-1
119
Figure 4-56 500rpm to 750rpm speed reference, motor speed estimate under
accelerating load-2
Figure 4-57 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-1
120
Figure 4-58 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-2
It can be deduced from Figure 4-53 to Figure 4-58 that the acceleration under
loading could be achieved by using Kalman filter state observer. The system noise is
corrected to some extent but the speed estimator performance under loading is worse
than the speed estimation performance speed adaptive flux observer.
4.1.4.4. The Kalman Filter for Speed Estimation Performance under No-
Load Speed Reversal
The speed estimator performance of the Kalman filter state observer is observed
under no-load speed reversal. The aim of this section is to determine the speed
estimator performance at zero speed crossing and low speed range.
Speed command, real rotor speed and estimated rotor speed for 100rpm to -
100rpm, 500 rpm to -500 rpm and 1000 rpm to -1000 rpm cases are represented at
Figure 4-31, Figure 4-32 and Figure 4-33 respectively.
121
Figure 4-59 100 rpm to -100 rpm speed reference, motor speed estimate under no-
load speed reversal
Figure 4-60 500 rpm to - 500 rpm speed reference, motor speed estimate under no-
load speed reversal
122
Figure 4-61 1000 rpm to -1000 rpm speed reference, motor speed estimate under no-
load speed reversal
As it can be seen from Figure 4-59, Figure 4-60 and Figure 4-61 both at low speeds
and the zero speed crossing, system performance decreases. It is diffucult to
conclude whether speed adaptive flux observer or Kalman filter for speed estimation
has better estimation performance at zero speed crossing.
The experiments of Kalman filter for speed estimation show that speed estimation
based on Kalman filter for speed estimation has limited tracking capability for whole
speed range and for no-load and with load cases. In general, Kalman filter state
observer has no superiority with respect to speed adaptive flux observer.
The main reason for poorer speed estimation performance to occur is that adaptive
scheme is internally corrected by speed estimation of flux observer; there is no
coupling between speed adaptive flux estimation scheme and Kalman filter for speed
estimation. So, closed-loop adaptive scheme is broken.
123
4.1.5 Experimental Results of Parallel Run of Speed Adaptive Flux
Observer and Kalman Filter for Speed Estimation
The induction motor parameters at Table 4-2 is used at the experimental stage of
both parallel run of speed adaptive flux observer and Kalman filter for speed
estimation.
In these experiments, motor is run in the closed- loop speed mode and the
quadrature encoder coupled to the shaft of the motor is utilized in order to verify the
estimated speed. The output of speed adaptive flux observer is utilized as speed
feedback. Also, the Kalman filter for speed estimation is running in order to verify its
performance when the adaptive scheme is enabled.
Both speed estimators designed with speed adaptive flux observer have been tested
experimentally for satisfactory operation at different speeds of motor.
The control parameters used at the speed adaptive flux observer experiments are
listed at Table 4-4
4.1.5.1. No-Load Experiments of Parallel Run of Speed Adaptive Flux
Observer and Kalman Filter for Speed Estimation
In these no-load experiments, motor is run in the closed- loop speed mode and the
quadrature encoder coupled to the shaft of the motor is utilized in order to verify the
estimated speed. The output of speed adaptive flux observer is utilized as speed
feedback.
Log of the mechanical rotor angle, the estimated speed, and the actual speed are
taken for 500rpm, 1000rp and, 1500rpm, constant speed request. The speed estimate
derived from actual rotor (quadrature encoder) position is given as dotted line.
124
Figure 4-62 500 rpm speed reference, motor speed estimate
Figure 4-63 1000 rpm speed reference, motor speed estimate
125
Figure 4-64 1500 rpm speed reference, motor speed estimate
It can be seen from Figure 4-62 to Figure 4-64 that no-load speed estimation
performance of Kalman filter state observer is enhanced. It is observed that the
measured percentage speed estimator error of Kalman filter state observer is less than
1.5 %. At steady-state, there is a speed offset about 10 rpm with respect to Adaptive
state observer. When adaptive scheme is enabled, the performance of Kalman state
observer is improved.
4.1.5.2. Parallel Run of Speed Adaptive Flux Observer and Kalman Filter for
Speed Estimation under Switched Loading
This section gives a comparative the performance analysis of the Kalman state
observer under switched loading while the output of speed adaptive flux observer
being utilized as speed feedback. The loading is obtained by using the Magtrol
dynamometer coupled to the shaft of the induction motor.
126
Speed command, real rotor speed and estimated rotor speed for 500 rpm 1000 rpm
and 1500 rpm cases are given in Figure 4-65, Figure 4-67, andFigure 4-69,
respectively. The load profiles are added to speed response at Figure 4-66, Figure
4-68 and Figure 4-70.
Figure 4-65 500 rpm speed reference, motor speed estimate under switched loading-1
127
Figure 4-66 500 rpm speed reference, motor speed estimate under switched loading-2
Figure 4-67 1000 rpm speed reference, motor speed estimate under switched loading-
1
128
Figure 4-68 1000 rpm speed reference, motor speed estimate under switched loading-
2
Figure 4-69 1500 rpm speed reference, motor speed estimate under switched loading-
1
129
Figure 4-70 1500 rpm speed reference, motor speed estimate under switched loading-
2
The experiments in this section show that when adaptive scheme is enabled, the
performance of Kalman state observer is improved. Tracking capability for whole
speed range and for no-load and with load cases are getting better. It is observed that
both at no-load and load cases, there is a speed offset about 10 rpm which worsen
steady-state performance.
4.1.5.3. Parallel Run of Speed Adaptive Flux Observer and Kalman Filter for
Speed Estimation under Accelerating Load
The speed estimator performance of the Kalman filter state observer is investigated
under accelerating torque. The aim of this section is to ensure sensorless vector drive
performance while accelerating load while the output of speed adaptive flux observer
is utilized as speed feedback.
130
Kalman filter for speed estimation model does not include dynamics of system.
Although this corresponds to infinite inertia, actually this is not true, but the required
correction is performed by the Kalman filter by the system noise, which also takes
account of the computational inaccuracies.
Speed command, real rotor speed and estimated rotor speed for 500 rpm to 750rpm
and 1000 to 1250 rpm cases are represented at Figure 4-71 and Figure 4-73
respectively. The load profiles are added to speed responses in Figure 4-72 and
Figure 4-74.
Figure 4-71 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-1
131
Figure 4-72 500 rpm to 750 rpm speed reference, motor speed estimate under
accelerating load-2
Figure 4-73 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-1
132
Figure 4-74 1000 rpm to 1250 rpm speed reference, motor speed estimate under
accelerating load-2
It can be deduced from Figure 4-71 and Figure 4-73 that the acceleration under
loading could be achieved by using Kalman filter state observer. The system noise is
corrected to some extent, but the speed estimator performance under loading is worse
than the speed estimation performance speed adaptive flux observer.
4.1.5.4. Parallel Run of Speed Adaptive Flux Observer and Kalman Filter for
Speed Estimation under No-load Speed Reversal
The speed estimator performance of Kalman filter state observer is observed under
no-load speed reversal. The aim of this section is to determine the speed estimator
performance at zero speed crossing and low speed range.
133
Speed command, real rotor speeds and estimated rotor speed for, 500 rpm to -500
rpm and 1000 rpm to -1000 rpm cases are given at Figure 4-75 and Figure 4-76.
Figure 4-75 500 rpm to -500 rpm speed reference, motor speed estimate under no-
load speed reversal
134
Figure 4-76 1000 rpm to -1000 rpm speed reference, motor speed estimate under no-
load speed reversal
4.2 Simulations of Extended Kalman Filter
Although, the low speed estimation performance of EKF is focused throughout
study, due to intense calculations in real-time in order to determine EKF states such
that rotor flux and stator currents and rotor speed, some overloading problems
occurred while performing real time EKF experiments on DSP. So, simulations of
EFK are performed. It is deduced from EKF simulations that EKF is convenient for
low speed operation and its performance could be increased by compansating voltage
errors caused by dead-time effects inverter switches, voltage drop in the power
electronic devices and the fluctuations of the dc link line voltage. Unfortunatelly,
those effects are not compansated at experiments and simulations for all the
estimation algorithms.
Simulations were performed in order to investigate effectiveness of the derived
algorithms for the extended Kalman filter (EKF) observer and to tune the covariance
matrices of it. The EKF observer is tuned by optimizing the entries of the
135
measurement noise covariance R matrix and the process noise covariance Q matrix.
In order to obtain fast and dynamic response and estimation accuracy MATLAB
Simulink is used as simulation tool. Convariance matrices, Q and R, are tuned by the
help of Matlab Response Optimization Toolbox.
The inputs of the EKF simulations are reconstructed phase voltages and logged
phase current outputs of the induction machine. Log of the mechanical rotor angle,
reconstructed phase voltages and phase currents are are taken from various
experiments in order to use them at simulations. Motor parameters at Table 4-2 are
used at the simulations of EKF observer.
The simulations are realized using phase voltages and phase current data obtained
from drive system by closed-loop speed control with encoder speed.
4.2.1. Tuning of EKF Covariance Matrices
In order to tune the covariance matrices of EKF, Matlab Response Optimization
Toolbox is used. The encoder, stationary reference frame variables isds, is
qs, Vsds and
Vsqs are logged from real-time experiments. The encoder signal is set as desired
waveform and entries of Q and R matrices are selected as tuned parameters, multiple
trials are performed to get an optimized speed estimator performance. Especially,
performance at lower speeds is focused. Thus, EKF speed estimatator experiments at
50 rpm, 100 rpm and 150 rpm are investigated.
4.2.2. Simulations at Lower Speeds
In this section, the performance of EKF is simulated under switched loading while
encoder is utilized as speed feedback. The loading is obtained by using the Magtrol
dynamometer coupled to the shaft of the induction motor. Encoder, stationary
reference frame isds, is
qs, Vsds and Vs
qs variables are logged from real-time
experiments and they are inputted to EKF simulation.
136
Real rotor speed and estimated rotor speed are represented for 50 rpm, 100 rpm,
and 150 rpm cases
Figure 4-77 50 rpm speed reference, motor speed estimate under switched loading
137
Since EKF estimates the stationary reference frame isds, and is
qs currents. Real and
Estimated stationary reference frame currents are given at Figure 4-78 and Figure
4-79 for 50 rpm speed command.
Figure 4-78 50 rpm speed reference, stationary reference frame isds current
138
Figure 4-79 50 rpm speed reference, stationary reference frame, isqs current
As it can be seen from Figure 4-77, the speed estimation performance of EKF is
better than speed adaptive state observer and Kalman filter state observer. The EKF
speed output at 50 rpm is oscillating which is due to some noise on stationary
reference frame voltages Vsds and Vs
qs. Since voltages are low, noise on voltages
become dominating on speed estimation.
The estimated isds and is
qs currents are exactly matched at the simulation for 50 rpm
speed command.
139
Figure 4-80 100 rpm speed reference, motor speed estimate under switched loading
Figure 4-81 100 rpm speed reference, stationary reference frame, isds current
140
Figure 4-82 100 rpm speed reference, stationary reference frame, isqs current
Figure 4-83 150 rpm speed reference, motor speed estimate under switched loading
141
Figure 4-84 150 rpm speed reference, stationary reference frame, isds current
Figure 4-85 150 rpm speed reference, stationary reference frame, isqs current
142
As it can be seen from Figure 4-77 - Figure 4-85 that the estimation performance at
low speed is better than the other two methods. The magnitute and frequency of
oscillations on speed estimate decreased when speed command is increased. In other
words, noise on the voltage terms becomes negligible when speed is increased
The estimates of current components are very well matched with direct and
quadrature axis currents derived with measured current terms.
143
CHAPTER 5
This work mainly includes implementation and experimental investigation of flux
and speed estimators for sensorless closed-loop speed control of induction motor.
The performance of observers is investigated in terms of steady-state and dynamic
speed response. The performance of speed estimators are compared based on
experimantal results and simulations.
Induction motor model based speed adaptive flux observer, Kalman filter state
observer and induction motor model based extended Kalman filter are implemented
throughout the study. In order to provide coherent control structure, mathematical
model of the induction motor has been derived both in stationary and synchronously
rotating dq axes system. In addition to that, space vector PWM and field orientation
concepts are introduced at the control system.
The response of the system against step loading has been tested on an experimental
set-up. Closed-loop speed control is enabled by utilizing closed-loop speed feedback
of sensorless speed estimation The test results are satisfactory in terms of accurary of
speed estimation and speed sensorless closed-loop vector control.
It has been deduced that adaptive state observer can be used for both speed
estimation and rotor flux estimation and it has ability to adapt itself to its speed
parameter which is estimated internally. Its speed estimation performance is
investigated under no-load, constant load and switched load cases. It performed well
at motor speeds greater than 100 rpm in terms of steady-state and dynamic
behaviour. The percentage speed estimation accuracy is measured lower than 1% at
speeds greater than 100 rpm. However, adaptive state observer is affected by the
motor parameter variations at 50 rpm and 100 rpm. The results obtained from the
adaptive speed observer tests are comparative to those reported in the literature
5 CONCLUSION
144
A Kalman filter state observer has been considered as the speed estimation scheme
for the motor speed in the study. The application requires the use of another
estimator for the estimation of the flux components and the rotor flux angle. So, The
output performance of the Kalman filter state observer is linked with the accuracy of
flux estimation and accuracy of speed observer. Complementary observer has been
chosen as speed adaptive flux observer algorithm. Since, speed adaptive flux
observer algorithm has ability to adapt itself to its speed parameter which is
estimated internally, flux observer becomes independent of closed-loop speed
feedback which leads to poorer performance compared with the estimated speed
output of speed adaptive flux observer. Closed-loop speed control based on Kalman
filter state observer percentage speed estimation accuracy is measured lower than 3%
at speeds greater than 250 rpm. It is observed that steady-state and dynamic
performance of Kalman filter state observer is poorer than speed adaptive state
observer. Also, it is more complex to implement it compared to speed adaptive state
observer.
Another estimator considered in the thesis work is the extented Kalman filter
(EKF) algorithm. The algorithm is introduced in the form of computer simulations
and script code for DSP implementation. The algorithm, however, could not have
been experimentally tested in the study. It is deduced from EKF simulations that
EKF is convenient for low speed operation and its performance could be improved
by compensating; voltage errors caused by dead-time effects in inverter switches,
voltage drop in the power electronic devices, and the fluctuations of the dc link line
voltage. The covariance matrices are optimized in order to ensure satisfactory low
speed performance by computer simulations. The percentage speed estimation
accuracy is calculated that EKF has estimated the rotor speed with ≤5% for the
speeds 50 rpm, 100 rpm and 150 rpm at steady state. When EKF simulation results
are compared with the results of the two other methods, EKF is convenient for low
speed operation.
145
The simulations and experiments conducted in the study have illustrated that it is
possible to increase the performance at low speeds at the expense of increased
computational burden on the processor. However, in order to control the motor at
zero speed, other techniques such as high frequency signal injection technique may
probably be used as well as a different electronic hardware.
For future work, the estimation accuracy and the dynamic response of the
estimators may be improved by compansating voltage errors caused by dead-time
effects inverter switches, voltage drop in the power electronic devices and the
fluctuations of the dc link line voltage. Also variations of stator and rotor resistances
could be modelled.
The estimators can be designed by other techniques, such as, extended Kalman
filter (EKF) technique, neural networks based estimators, sliding mode estimators.
Furthermore, more advanced control structures can be investigated for better control
of both motor current loop and speed loop. Some hardware upgrades together with
high frequency signal implementation could be done to control the motor at zero
speed.
146
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